you can connect to the database using SqlConnection and SqlCommand and execute the following command text for example:

BACKUP DATABASE [MyDatabase] TO  DISK = 'C:\....\MyDatabase.bak'

See here for examples.

It's a good practice to use a config file like this:

<?xml version="1.0" encoding="utf-8"?>
<configuration>
  <connectionStrings>
    <add name="MyConnString" connectionString="Data Source=(local);Initial Catalog=MyDB; Integrated Security=SSPI" ;Timeout=30"/>
  </connectionStrings>
  <appSettings>
    <add key="BackupFolder" value="C:/temp/"/>
  </appSettings>
</configuration> 

Your C# code will be something like this (note the parameterization of the SQL):

// read connectionstring from config file
var connectionString = ConfigurationManager.ConnectionStrings["MyConnString"].ConnectionString; 

// read backup folder from config file ("C:/temp/")
var backupFolder = ConfigurationManager.AppSettings["BackupFolder"];
 
var sqlConStrBuilder = new SqlConnectionStringBuilder(connectionString);

// set backupfilename (you will get something like: "C:/temp/MyDatabase-2013-12-07.bak")
var backupFileName = String.Format("{0}{1}-{2}.bak", 
    backupFolder, sqlConStrBuilder.InitialCatalog, 
    DateTime.Now.ToString("yyyy-MM-dd"));

const string query = @"
BACKUP DATABASE @db
TO DISK = @file;
";
using (var connection = new SqlConnection(sqlConStrBuilder.ConnectionString))
using (var command = new SqlCommand(query, connection))
{
    command.Parameters.Add("@db", SqlDbType.NVarChar, 128).Value = sqlConStrBuilder.InitialCatalog;
    command.Parameters.Add("@file", SqlDbType.NVarChar, 255).Value = backupFileName;

    connection.Open();
    command.ExecuteNonQuery();
}

Works for me:

public class BackupService
{
    private readonly string _connectionString;
    private readonly string _backupFolderFullPath;
    private readonly string[] _systemDatabaseNames = { "master", "tempdb", "model", "msdb" };

    public BackupService(string connectionString, string backupFolderFullPath)
    {
        _connectionString = connectionString;
        _backupFolderFullPath = backupFolderFullPath;
    }

    public void BackupAllUserDatabases()
    {
        foreach (string databaseName in GetAllUserDatabases())
        {
            BackupDatabase(databaseName);
        }
    }

    public void BackupDatabase(string databaseName)
    {
        string filePath = BuildBackupPathWithFilename(databaseName);

        using (var connection = new SqlConnection(_connectionString))
        {
            var query = String.Format("BACKUP DATABASE [{0}] TO DISK='{1}'", databaseName, filePath);

            using (var command = new SqlCommand(query, connection))
            {
                connection.Open();
                command.ExecuteNonQuery();
            }
        }
    }

    private IEnumerable<string> GetAllUserDatabases()
    {
        var databases = new List<String>();

        DataTable databasesTable;

        using (var connection = new SqlConnection(_connectionString))
        {
            connection.Open();

            databasesTable = connection.GetSchema("Databases");

            connection.Close();
        }

        foreach (DataRow row in databasesTable.Rows)
        {
            string databaseName = row["database_name"].ToString();

            if (_systemDatabaseNames.Contains(databaseName))
                continue;

            databases.Add(databaseName);
        }

        return databases;
    }

    private string BuildBackupPathWithFilename(string databaseName)
    {
        string filename = string.Format("{0}-{1}.bak", databaseName, DateTime.Now.ToString("yyyy-MM-dd"));

        return Path.Combine(_backupFolderFullPath, filename);
    }
}

            SqlConnection con = new SqlConnection();
            SqlCommand sqlcmd = new SqlCommand();
            SqlDataAdapter da = new SqlDataAdapter();
            DataTable dt = new DataTable();

            con.ConnectionString = ConfigurationManager.ConnectionStrings["MyConString"].ConnectionString;
            string backupDIR = "~/BackupDB";
            string path = Server.MapPath(backupDIR);

            try
            {
                var databaseName = "MyFirstDatabase";
                con.Open();
                string saveFileName = "HiteshBackup";
                sqlcmd = new SqlCommand("backup database" +databaseName.BKSDatabaseName + "to disk='" + path + "\\" + saveFileName + ".Bak'", con);
                sqlcmd.ExecuteNonQuery();
                con.Close();                 


                ViewBag.Success = "Backup database successfully";
                return View("Create");
            }
            catch (Exception ex)
            {
                ViewBag.Error = "Error Occured During DB backup process !<br>" + ex.ToString();
                return View("Create");
            }

I have new method without SMO problems

1. Create .bat File with backup sqlcmd command

for backup

SqlCmd -E -S Server_Name –Q “BACKUP DATABASE [Name_of_Database] TO DISK=’X:PathToBackupLocation[Name_of_Database].bak'”

for restore

SqlCmd -E -S Server_Name –Q “RESTORE DATABASE [Name_of_Database] FROM DISK=’X:PathToBackupFile[File_Name].bak'”

2. Run the the bat file with WPF/C# code

        FileInfo file = new FileInfo("DB\\batfile.bat");
        Process process = new Process();
        process.StartInfo.FileName = file.FullName;
        process.StartInfo.Arguments = @"-X";
        process.StartInfo.WindowStyle = ProcessWindowStyle.Normal;
        process.StartInfo.UseShellExecute = false; //Changed Line
        process.StartInfo.RedirectStandardOutput = true;  //Changed Line
        process.Start();
        string output = process.StandardOutput.ReadToEnd(); //Changed Line
        process.WaitForExit(); //Moved Line

You can take database back-up of SQL server instance using C#, as below

Step 1: Install Nuget package "Install-Package Microsoft.SqlServer.SqlManagementObjects"

Step 2: Use the below C# Command to take backup using Custom function

public void BackupDatabase(string databaseName, string userName, string password, string serverName, string destinationPath)

 {  
//Define a Backup object variable.
Backup sqlBackup = new Backup();
//Specify the type of backup, the description, the name, and the database to be backed up.
sqlBackup.Action = BackupActionType.Database;

sqlBackup.BackupSetDescription = "BackUp of:" + databaseName + "on" + DateTime.Now.ToShortDateString();

sqlBackup.BackupSetName = "FullBackUp";

sqlBackup.Database = databaseName;
//Declare a BackupDeviceItem
BackupDeviceItem deviceItem = new BackupDeviceItem(destinationPath + "FullBackUp.bak", DeviceType.File);

//Define Server connection
ServerConnection connection = new ServerConnection(serverName, userName, password); //To Avoid TimeOut Exception
Server sqlServer = new Server(connection);
sqlServer.ConnectionContext.StatementTimeout = 60 * 60;
Database db = sqlServer.Databases[databaseName];
// Reference Database As microsoft.sqlserver.management.smo.database, not as System.entity.database

sqlBackup.Initialize = true;

sqlBackup.Checksum = true;

sqlBackup.ContinueAfterError = true;
//Add the device to the Backup object.
sqlBackup.Devices.Add(deviceItem);

//Set the Incremental property to False to specify that this is a full database backup. 
sqlBackup.Incremental = false;
sqlBackup.ExpirationDate = DateTime.Now.AddDays(3);

//Specify that the log must be truncated after the backup is complete.        
sqlBackup.LogTruncation = BackupTruncateLogType.Truncate;
sqlBackup.FormatMedia = false;

//Run SqlBackup to perform the full database backup on the instance of SQL Server. 
sqlBackup.SqlBackup(sqlServer);

//Remove the backup device from the Backup object.           
 sqlBackup.Devices.Remove(deviceItem);

}

Add References

Microsoft.SqlServer.ConnectionInfo
Microsoft.SqlServer.Management.Sdk.Sfc
Microsoft.SqlServer.Smo
Microsoft.SqlServer.SmoExtended
Microsoft.SqlServer.SqlEnum

That's it, you are done, it will take backup of specified database at specified location passed to the function.

Source: Various ways to back up SQL server database

Note: User must have proper rights to write backup data on specified disk location.

This worked for me...

private void BackupButtonClick(object sender, RoutedEventArgs e)
{
    // FILE NAME WITH DATE DISTICNTION
    string fileName = string.Format("SchoolBackup_{0}.bak", DateTime.Now.ToString("yyyy_MM_dd_h_mm_tt"));
    try
    {
        // YOUR SEREVER OR MACHINE NAME
        Server dbServer = new Server (new ServerConnection("DESKTOP"));
        Microsoft.SqlServer.Management.Smo.Backup dbBackup = new Microsoft.SqlServer.Management.Smo.Backup()
        {
            Action = BackupActionType.Database, 
            Database = "School"
        };

        dbBackup.Devices.AddDevice(@backupDirectory() +"\\"+ fileName, DeviceType.File);
        dbBackup.Initialize = true;
        dbBackup.SqlBackupAsync(dbServer);


        MessageBox.Show("Backup", "Backup Completed!");
    }
    catch(Exception err)
    {
        System.Windows.MessageBox.Show(err.ToString());
    }
}


// THE DIRECTOTRY YOU WANT TO SAVE IN
public string backupDirectory()
{
    using (var dialog = new FolderBrowserDialog())
    {
        var result = dialog.ShowDialog();
        return dialog.SelectedPath;
    }
}

You can use the following queries to Backup and Restore, you must change the path for your backup

Database name=[data]

Backup:

BACKUP DATABASE [data] TO  DISK = N'C:\Program Files\Microsoft SQL Server\MSSQL10_50.MSSQLSERVER\MSSQL\Backup\data.bak' WITH NOFORMAT, NOINIT,  NAME = N'data-Full Database Backup', SKIP, NOREWIND, NOUNLOAD,  STATS = 10
GO

Restore:

RESTORE DATABASE [data] FROM  DISK = N'C:\Program Files\Microsoft SQL Server\MSSQL10_50.MSSQLSERVER\MSSQL\Backup\data.bak' WITH  FILE = 1,  NOUNLOAD,  REPLACE,  STATS = 10
GO

 private void BackupManager_Load(object sender, EventArgs e)
        {
            txtFileName.Text = "DB_Backup_" + DateTime.Now.ToString("dd-MMM-yy");
        }

        private void btnDBBackup_Click(object sender, EventArgs e)
        {
            if (!string.IsNullOrEmpty(txtFileName.Text.Trim()))
            {
                BackUp();
            }
            else
            {
                MessageBox.Show("Please Enter Backup File Name", "", MessageBoxButtons.OK, MessageBoxIcon.Information);
                txtFileName.Focus();
                return;
            }
        }

        private void BackUp()
        {
            try
            {

                progressBar1.Value = 0;

                for (progressBar1.Value = 0; progressBar1.Value < 100; progressBar1.Value++)
                {

                }

                pl.DbName = "Inventry";
                pl.Path = @"D:/" + txtFileName.Text.Trim() + ".bak";

                for (progressBar1.Value = 100; progressBar1.Value < 200; progressBar1.Value++)
                {

                }

                bl.DbBackUp(pl);
                for (progressBar1.Value = 200; progressBar1.Value < 300; progressBar1.Value++)
                {

                }

                for (progressBar1.Value = 300; progressBar1.Value < 400; progressBar1.Value++)
                {

                }

                for (progressBar1.Value = 400; progressBar1.Value < progressBar1.Maximum; progressBar1.Value++)
                {

                }
                if (progressBar1.Value == progressBar1.Maximum)
                {
                    MessageBox.Show("Backup Saved Successfully...!!!", "", MessageBoxButtons.OK, MessageBoxIcon.Information);
                }
                else
                {
                    MessageBox.Show("Action Failed, Please try again later", "", MessageBoxButtons.OK, MessageBoxIcon.Error);
                }
            }

            catch (Exception ex)
            {
                MessageBox.Show("Action Failed, Please try again later", "", MessageBoxButtons.OK, MessageBoxIcon.Error);
            }
            finally
            {
                progressBar1.Value = 0;
            }
        }