Here is an example XNA 4.0 program that draws a simple textured square. It requires the Green-gel-x texture (from wiki-commons - link in code) added to the content project (or replaced with your own texture). After the textured square is drawn, a wireframe square is drawn over the top so you can see the triangles. This example uses an orthographic projection and a BasicEffect instead of an effect file but is otherwise similar to the Riemer tutorial you linked to.

To perform the texturing each vertex requires a texture coordinate. For a texture that tiles once across the surface of the square the texture coordinates are (0, 0) for the top left vertex and (1, 1) for the bottom right vertex and so on. If you wanted to tile the texture across the square twice, you could set all the bottom right texture coordinates to 2, instead of 1.

using Microsoft.Xna.Framework;
using Microsoft.Xna.Framework.Graphics;
using Microsoft.Xna.Framework.Input;

namespace WindowsGame
{
    public class Game1 : Microsoft.Xna.Framework.Game
    {
        const string TEXTURE_NAME = "Green-gel-x";  // http://upload.wikimedia.org/wikipedia/commons/9/99/Green-gel-x.png
        const int TOP_LEFT = 0;
        const int TOP_RIGHT = 1;
        const int BOTTOM_RIGHT = 2;
        const int BOTTOM_LEFT = 3;
        RasterizerState WIREFRAME_RASTERIZER_STATE = new RasterizerState() { CullMode = CullMode.None, FillMode = FillMode.WireFrame };

        GraphicsDeviceManager graphics;
        BasicEffect effect;
        Texture2D texture;
        VertexPositionColorTexture[] vertexData;
        int[] indexData;
        Matrix viewMatrix;
        Matrix projectionMatrix;

        public Game1()
        {
            graphics = new GraphicsDeviceManager(this);
            Content.RootDirectory = "Content";
        }

        protected override void Initialize()
        {
            effect = new BasicEffect(graphics.GraphicsDevice);

            SetUpVertices(Color.White);
            SetUpCamera();
            SetUpIndices();

            base.Initialize();
        }

        private void SetUpVertices(Color color)
        {
            const float HALF_SIDE = 200.0f;
            const float Z = 0.0f;

            vertexData = new VertexPositionColorTexture[4];
            vertexData[TOP_LEFT] = new VertexPositionColorTexture(new Vector3(-HALF_SIDE, HALF_SIDE, Z), color, new Vector2(0, 0));
            vertexData[TOP_RIGHT] = new VertexPositionColorTexture(new Vector3(HALF_SIDE, HALF_SIDE, Z), color, new Vector2(1, 0));
            vertexData[BOTTOM_RIGHT] = new VertexPositionColorTexture(new Vector3(HALF_SIDE, -HALF_SIDE, Z), color, new Vector2(1, 1));
            vertexData[BOTTOM_LEFT] = new VertexPositionColorTexture(new Vector3(-HALF_SIDE, -HALF_SIDE, Z), color, new Vector2(0, 1));
        }

        private void SetUpIndices()
        {
            indexData = new int[6];
            indexData[0] = TOP_LEFT;
            indexData[1] = BOTTOM_RIGHT;
            indexData[2] = BOTTOM_LEFT;

            indexData[3] = TOP_LEFT;
            indexData[4] = TOP_RIGHT;
            indexData[5] = BOTTOM_RIGHT;
        }

        private void SetUpCamera()
        {
            viewMatrix = Matrix.Identity;
            projectionMatrix = Matrix.CreateOrthographic(Window.ClientBounds.Width, Window.ClientBounds.Height, -1.0f, 1.0f);
        }

        protected override void LoadContent()
        {
            texture = Content.Load<Texture2D>(TEXTURE_NAME);
        }

        protected override void Update(GameTime gameTime)
        {
            if (GamePad.GetState(PlayerIndex.One).Buttons.Back == ButtonState.Pressed)
                this.Exit();

            base.Update(gameTime);
        }

        protected override void Draw(GameTime gameTime)
        {
            GraphicsDevice.Clear(Color.CornflowerBlue);

            // Draw textured box
            GraphicsDevice.RasterizerState = RasterizerState.CullNone;  // vertex order doesn t matter
            GraphicsDevice.BlendState = BlendState.NonPremultiplied;    // use alpha blending
            GraphicsDevice.DepthStencilState = DepthStencilState.None;  // don t bother with the depth/stencil buffer

            effect.View = viewMatrix;
            effect.Projection = projectionMatrix;
            effect.Texture = texture;
            effect.TextureEnabled = true;
            effect.DiffuseColor = Color.White.ToVector3();
            effect.CurrentTechnique.Passes[0].Apply();

            GraphicsDevice.DrawUserIndexedPrimitives(PrimitiveType.TriangleList, vertexData, 0, 4, indexData, 0, 2);

            // Draw wireframe box
            GraphicsDevice.RasterizerState = WIREFRAME_RASTERIZER_STATE;    // draw in wireframe
            GraphicsDevice.BlendState = BlendState.Opaque;                  // no alpha this time

            effect.TextureEnabled = false;
            effect.DiffuseColor = Color.Black.ToVector3();
            effect.CurrentTechnique.Passes[0].Apply();

            GraphicsDevice.DrawUserIndexedPrimitives(PrimitiveType.TriangleList, vertexData, 0, 4, indexData, 0, 2);

            base.Draw(gameTime);
        }
    }
}