Why isn't the hot air leaving my apartment even with a window open in the bedroom and living room wide open?

You need some sort of circulation otherwise the hot air is just going to sit there. My room has horrible circulation so what I did was set up one fan my my window blowing out and another by the other window blowing in. This helped pull cool air from outside into my room and push the hot air out.

Yes, I approached this problem from a nerdy standpoint thinking of my room as a big computer :P

This is why they invented window box fans. For $20 you can put one of those noisy units in a window on the other side of the house. First, light a match near the window and blow it out. Watch the direction of the smoke. Which way did it go – in or out? Then set the fan up to blow in that same direction. Then turn it on.

I have an indoor / outdoor remote thermometer. When it is cooler outdoors I turn on the fans. As soon as it gets hotter outdoors than inside I turn it off.

in my window*...

In the window if you have the raise to open type windows. That is what box fans are made for. With windows which open sideways, and have a track instead of a windowsill, it is better to have an oscillating floor fan, and set it near the window.
Ooooooooooooooor, go to the library and enjoy the air conditioning your taxes paid for.

I too live upstairs and have the same problem. It’s worse in hot weather though. I have a box ventilation fan in the kitchen and bathroom, but nothing in the other rooms. On hot days, the living room, even with both windows open is like an oven. More especially so as the design of my windows won’t allow me to fully open them. So I recently purchased a standing floor fan that’s large enough to provide circulating airflow round the whole room, while standing somewhere out of the way. There are 3 power levels – even at it’s lowest setting it’s like sitting in a cool breeze. I wouldn’t be without my fan now, and wish I’d bought one years ago!

Look up Bernouilli Effect in wikipedia. There has to be some pressure difference to cause the change in movement of air.

Hot air rises and cold air sinks, get a ceiling fan to mix it up and a window box fan to blow it out/in or suck it out/in.

If you have double hung windows, pull one down from the top, and one up from the sill. You should get a little flow as the hot air rises and escapes, and the cool air replaces it. Needless to say, a fan blowing into the room will speed it up.

You can buy floor standing AC units now, too.

If you open windows on opposite ends of the apartment (or even caddy corner to each other) and you have a window fan (like @LuckyGuy described) in one window pulling air out, the air will flow really nicely through your place. Try that.