Once you have created a cool looking primitive, you can apply the edit poly modifier via the create tab. This enables you to modify the vertices, edges, and faces of the object individually and do other cool stuff to them. Here are some of the basic stuff.
Chamfer: This turns a Vertex into 3 that slide along edges:
Extrude: Vertex extrude is similar to chamfer that it creates 3 vertices that can slide along edges, but a 4th vertex is creates that extends out from the mesh.
Target Weld: This allows you to weld one vertex to another vertex that is connected to it via an edge. This Will remove the vertex that is being welded and attach the edges to the vertex that it was welded to. Useful for optimising a model and removing unused vertices.
Remove: This allows you to remove a vertex. unlike just deleting one, the hole that is left is automatically filled with a polygon.
Break: This turns one vertex into multiple, one for each poly.
Connect: This is important. Connect will add vertices and edges between two edges, connecting them together.
Polygon Editing. (Fun ones)
Extrude: This allows you to pull a face out of a face, or push a face into a face. Commonly used.
Bevel: Similar to extrude, but lets you change the size of the face you are extruding.
Inset: Creates a face inside a face.
Bridge: Connects a face to a face, connecting edges between them.