As the first 80+ degree day ends in Chicago, I sit here in my top-floor apartment and just remembering how damn hot it gets in here during the summer. Now that it’s dark out, I want to basically figure out the most effective way of getting outside air into my apartment to cool it down a bit.
I’ve known people who pointed the fan inward, so that the fan sits by the window and drags in outside air. But I’ve also known people who swear by the practice of pointing the fan to the window, I’m assuming so that the warmer air inside goes outside and outside air gets sucked in.
Which is more effective?
The high school I went to was three stories, brick, and NOT air conditioned. Above the door of each classroom was a huge fan. The most common thing was to open the windows (six huge windows per class opposite the door) all the way. This put a strain on the fan trying to pull all that air.
One teacher got it right: open each window about two inches and set the fan to blow out into the hallways. Within 15 minutes we were begging her to shut it off.
So, I would suggest setting the fan to exhaust and open the other windows (assuming you have more than one) about a couple of inches.
You only have one window in your apartment?
Otherwise, if you have at least two windows, which ideally are in separate rooms, what you do is put the fan in one window and point it outward, so it’s sucking cool outside air in through the other window.
And in my personal experience of Downstate Illinois summers, the best window to have the cool air being sucked in is the bedroom, 'cause the room with the cool air being sucked in will cool off faster than the room with the fan blowing outwards, so put the fan blowing outwards in the living room or kitchen.
And, you wanna arrange your airflow. Get a big box fan and wedge it in the window so the only place it can draw air from is from the rest of the room behind it and thus from the other room, and thus it’s forced to draw the cool incoming air from the other window–IOW, don’t waste your time with one of those little stand fans sitting on a table in front of the window or something, because then it can draw air in from the hot outside air that’s directly in front of it.
You can also close closets (and the bathroom) to further force the fan to draw air only from the other window.
But if you only have one window, buy an air conditioner. 
Picture a map of your apartment. Generally, it’s best to go with the prevailing wind. If it’s toward the east, for example, put an exhaust fan on one or two of the east windows. Air will choose the path of least resistance, so you may have to close the middle windows, if any, to make sure you get airflow all the way through. Most box fans are roughly 20" by 20", so set each fan over to one side of its windowsill. Cut a piece of a box to block the gap, and lay it up against the screen. The windiest place in the joint will probably be in the hallway. Sit there with a big lemonade.
I do have a good straight-line air flow in my apartment (window on each side - one being a normal window and the other being a sliding glass door to the balcony) that faces East-West. Unfortunately, my apartment building is inbetween two other buildings of the exact same design (i.e. same height), so there is suprisingly little wind that comes through the windows. I’m gonna have to get pretty clever to harness whatever wind we do get 
Thanks for the advice, I think I’m gonna pick me up a box fan here soon.
I would make an exception for a window nearest cooler air. For instance if one window opened into a shaded area I would use that as the intake window.