A flat is an apartment. If a girl invites you up to her flat, she's not trying to squash you, she's inviting you over. And she's probably from England, because flat is much more common in British than American English.
Geometrically, something flat is an even, level plane, like Kansas relative to the Rocky Mountains, or the cargo area of a flatbed truck. As an adjective, flat means "less than lively." A flat soda has no bubbles. If your speech falls flat, no one was excited. If you get mugged you might choose to lie flat on the ground. A flat note is just below the right pitch. And if you're cooking is flat, it's boring.