How do UVs work?

UV(W) coordinates are a normalized (0 to 1) 2-Dimensional coordinate system, where the origin (0,0) exists in the bottom left corner of the space. UV is in this case synonymous to XY, but since XY is used in the XYZ 3D space so we use UVW in the texture space to avoid confusion.

Note that there is a W which corresponds to the Z axis, but it won't be used in a Texture2D as it is 2D.

So with that said, think of this 0 to 1 space as 0% to 100% of the image. If your texture is 128x128 then having a coordinate of 1 in either the U or V would mean that that particular vertex will exist at the normalized bounds of our texture space.

Example: a vertex with a UV of .5,.5 will be in the exact center of the image.

This works for non-square as well. Example: an image of 512,128 and a vertex with a UV coordinate of .5,.5 will exist at 256,64 in our texture.

Now, you asked how a 4vert quad would look, in short it would look something like this:

top-left 0,1 top-right 1,1 bottom-left 0,0 bottom-right 1,0

Depending on how you've assembled your triangles will vary on what vertex is what, but follow those guidelines and you should be able to figure it out.

[Edit] Here is a quick class that builds a quad. I did this step by step and tried to avoid a lot of the short hand I'd normally use to be as decipherable as possible.

 using UnityEngine;
 using System.Collections;

 [RequireComponent(typeof(MeshFilter),typeof(MeshRenderer))]
 public class DrawQuad : MonoBehaviour 
 {

     // Use this for initialization
     void Start ()
     {
         Mesh mesh = new Mesh();
     
         Vector3[] vertices = new Vector3[4];
         vertices[0] = new Vector3(0,0,0); //top-left
         vertices[1] = new Vector3(1,0,0); //top-right
         vertices[2] = new Vector3(0,-1,0); //bottom-left
         vertices[3] = new Vector3(1,-1,0); //bottom-right
         
         mesh.vertices = vertices;
         
         int[] triangles = new int[6]{0,1,2,3,2,1};
         mesh.triangles = triangles;
         
         //this is also acceptable!
         //mesh.SetTriangleStrip(new int[4]{0,1,2,3}, 0);
     
         Vector2[] uvs = new Vector2[4];
         uvs[0] = new Vector2(0,1); //top-left
         uvs[1] = new Vector2(1,1); //top-right
         uvs[2] = new Vector2(0,0); //bottom-left
         uvs[3] = new Vector2(1,0); //bottom-right
     
         mesh.uv = uvs;
     
         Vector3[] normals = new Vector3[4]{Vector3.forward,Vector3.forward,Vector3.forward,Vector3.forward};
         mesh.normals = normals;
     
         //you could also call this instead...
         //mesh.RecalculateNormals();
     
     
         //grab our filter.. set the mesh
         MeshFilter filter = GetComponent<MeshFilter>();
         filter.mesh = mesh;
     
         //you can do your material stuff here...
         //MeshRenderer r = GetComponent<MeshRenderer>();
         //r.material = new Material(Shader.Find("my_shader_here"));
     
     }
 }

Hope that helps.

==