I'll do my best to explain
Casual is where anything is allowed, but things might be frowned upon by your playgroup
Modern is where there are banned cards and sets after 8th edition allowed
Commander/EDH is where you can have only 1 of every card except basic lands, and you have a commander which is in the command zone. The only colors allowed are your commanders colors
Legacy is all sets allowed, but cards are banned
Vintage is where you can have 1 of the cards on the restricted list, and all sets allowed
Standard, atm anyways, is where you can only use certain sets, nothing banned from those sets. The current sets are Dragons of Tarkir, Origins, Battle of Zendikar, Oath of the Gatewatch, Shadows over Innistrad, and Eldritch Moon.
Does that help a little?