How can I create a two dimensional array that containts Ints and Strings in C#?

You should really learn some basics of working with databases. Your User class is something that is called Model and what you're looking for is collection.

First of all i'd recommend to add id field to make it easier to identify each object (though it's optional):

public class User
{
    public int id { get; set; }
    public string userName { get; set; }
    public string password { get; set; }
    public int savingsAcct { get; set; }
    public int checkAcct { get; set; }
}

And you can use List<User> :

List<User> users = new List<User>()
{
   new User() { id = 0, userName = "Alex", etc... }
   new User() { id = 1, userName = "Joshua", etc... }
   new User() { id = 2, userName = "Phil", etc... }
};

To find user with specific userName you'd have to use LINQ here :

//                returns null if no such user is found
User alex = users.FirstOrDefault(u => u.userName == "Alex");

Sometimes it's more suitable to use Dictionary<TKey, TValue> than List<T> :

Dictionary<string, User> users = new Dictionary<string, User>()
{
   { "Alex", new User() { id = 0, userName = "Alex", etc... } },
   { "Joshua", new User() { id = 1, userName = "Joshua", etc... } },
   { "Phil", new User() { id = 2, userName = "Phil", etc... } },
};

Then you can access users by their userName with help of indexer :

User alex = users["Alex"];

This shows a working example, but will need to be massaged into your code:

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace ConsoleApplication1
{
    public class User
    {
        public string userName { get; set; }
        public string password { get; set; }
        public int savingsAcct { get; set; }
        public int checkAcct { get; set; }
    }

    class Program
    {
        static void Main(string[] args)
        {
            // here you'll take the user and password from the form
            string userNameEnteredIntoScreen = "???";
            string passwordEnteredIntoScreen = "???";

            // build list of valid users (amend to suit)
            var users = new List<User>();
            users.Add(new User
            {
                userName = "user1",
                password = "abc",
                savingsAcct = 1,
                checkAcct = 2,
            });
            users.Add(new User
            {
                userName = "user2",
                password = "xyz",
                savingsAcct = 3,
                checkAcct = 4,
            });

            // then to check if they're valid
            var user = users.SingleOrDefault(u =>
                u.userName == userNameEnteredIntoScreen &&
                u.password == passwordEnteredIntoScreen
                );

            if (user != null)
            {
                // the user and password was valid, and 'user' variable
                // now contains the details of the user
            }
        }
    }
}

As clarified in the comments, you are looking for a collection that can store User objects in your application.

If you need to be able to change the contents of the collection, a List<T> is a good choice - it does not have a fixed size so it is easy to add and remove objects.

List<User> _allUsers = new List<User>();
_allUsers.Add(/* user object */); //populate the list with Users using .Add()

foreach (var user in _allUsers)
{
    if(user.userName == someInputVariable && user.passWord == someOtherInputVariable)
    {
        ...
    }
}

MSDN documentation for List<T>

You could also use a 1-dimensional array of User but I think a List<T> makes more sense for what you are trying to do, and it is very easy to work with as someone who is just starting to learn the language.

In C#, You cannot create a two dimensional array with two different data types, in your case, int and string. You can only create a two dimensional array of the same data type.

If you require a data structure to hold two data types, you can use a Dictionary pairs.

//e.g.
Dictionary<int, string> data = new Dictionary<int, string>
{ 
   {1, "value 1"},
   {2, "value 2"},
   {3, "value 3"}
};


//With your POCO can create a Dictionary as below
Dictionary<int, User> data = new Dictionary<int, User>
{ 
   {1, new User{ userName = "username1", password = "password1", savingsAcct  = 1,checkAcct = 1 }},
   {2, new User{ userName = "username2", password = "password2", savingsAcct  = 1,checkAcct = 2 }},
   {3, new User{ userName = "username3", password = "password3", savingsAcct  = 1,checkAcct = 3}}
};

You can use a loop to initialize your dictionary