How do I backup Sql Server Express database programatically?

You could have your application launch a .sql file that does the backing up or restoring for you. Granted, I've mostly seen this done in a Scheduled Task, but I suppose you could do it from a triggered event inside your app.

System.Diagnostics.Process p = new System.Diagnostics.Process();
p.StartInfo.FileName = "C:\\path\\to\\backup.sql";
p.Start();

UPDATE:

As pointed out in the comments, this won't work if you don't have SQL Management Studio installed on the server. Alternatively, you could call a stored procedure. Upon reflecting, I'm not sure why I didn't suggest the stored proc first - probably because the other methodology was fresh on my brain due to being forced to implement it that way in a previous project.

You can request a backup from within your app by executing:

@"BACKUP DATABASE [MyDBName] TO DISK = 'c:\somedir\MyDBName.bak' WITH INIT" (ref)

Or use SQL SMO Objects SqlBackup() directly.

You could use SQL Server Management Objects

First add a reference in your project to: Microsoft.SqlServer.Smo.dll, Microsoft.SqlServer.SmoExtended.dll, Microsoft.SqlServer.SqlEnum.dll and Microsoft.SqlServer.SmoEnum.dll.

After that to Backup your Database follow this sample:

//Connect to the server
Server srv = new Server();
//If Sql Server is not local or is a named instance you could do 
//Server srv = new Server("SERVERNAME");

Database db = srv.Databases("YourDB");

//Create a backup definition
Backup backup = new Backup(); 

backup.Action = BackupActionType.Database; 
backup.BackupSetDescription = "Full backup of Adventureworks2008R2"; 
backup.BackupSetName = "My app Backup"; 
backup.Database = "YourDB"; 

//Configure the backup device

BackupDeviceItem backupDevice = new BackupDeviceItem("YourDB.bak", DeviceType.File); 

backup.Devices.Add(backupDevice); 

//Specify wether do a full backup or not
backup.Incremental = false; 

//Specify log truncation mode
backup.LogTruncation = BackupTruncateLogType.Truncate; 

//Do the backùp
backup.SqlBackup(srv);