How do you make mesh in Unity?

You can create Meshes procedurally, which is useful if your custom shapes can be generated from code.

See http://unity3d.com/support/documentation/ScriptReference/Mesh.html

You can even add custom menu handlers to the editor so that your shapes can be added just like the standard ones.

For "artistic" shapes, +1 to the Blender suggestion, I wouldn't try and program anything more complicated than a teapot if I were you! :-)

Everything is about vertexes, triangles, edges, normals, UVs etc... Example:

 using UnityEngine;
 using System.Collections;
 
 public class Plane_Mesh_Script : MonoBehaviour {
 
     public Vector3[] Vertex = new Vector3[] 
     {
         new Vector3(0,0,0),new Vector3(1,0,0),new Vector3(0,1,0),new Vector3(1,1,0)
     };
 
     public Vector2[] UV_MaterialDisplay = new Vector2[] 
     {
         new Vector2(0,0),new Vector2(1,0),new Vector2(0,1),new Vector2(1,1) // 4 UV with all directions! (Plane has 4 uvMaps)
     };
 
     public int[] Triangles = new int[6]; // 2 Triangle combinations (2*3=6 vertices/vertexes)
 
     public Material material;
     
     void Start () {
 
         Triangles [0] = 0;
         Triangles [1] = 3;
         Triangles [2] = 1;
 
         Triangles [3] = 0;
         Triangles [4] = 2;
         Triangles [5] = 3;
 
         Mesh mesh = new Mesh ();
         transform.GetComponent<MeshFilter> ();
 
         if(!transform.GetComponent<MeshFilter> () ||  !transform.GetComponent<MeshRenderer> () ) //If you havent got any meshrenderer or filter
         {
             transform.gameObject.AddComponent<MeshFilter>();
             transform.gameObject.AddComponent<MeshRenderer>();
         }
 
         transform.GetComponent<MeshFilter> ().mesh = mesh;
 
         mesh.name = "MyOwnObject";
 
         mesh.vertices = Vertex;
         mesh.triangles = Triangles;
         mesh.uv = UV_MaterialDisplay;
 
         mesh.RecalculateNormals ();
         mesh.Optimize ();
         transform.gameObject.renderer.material = material; //If you want a material.. you have it :)
 
     }
 }

Unity is a game engine not a modeling program. So you cannot edit meshes in Unity.

If you are looking for a free modeler, Blender is always useful. Download 2.49 because 2.55 is still a beta, and didn't work well with Unity last time I checked.