removing sim card from iphone 4

A SIM unlock tool works best. I picked up a couple on eBay for $0.99. Alternatively, you could insert the end of a small paper clip into the hole in the SIM slot (on the right side of the phone) and it should slide out.

Here you go: http://support.apple.com/kb/HT5163

That's what I tried using a paper clip but nothing was sliding out. Is it supposed to pop out like a CD drive on a laptop? I did not want to poke the hole too hard and damage it. Is the pin supposed to go in straight at a right angle, or is it supposed to go in at a slight angle to "hook" the tray out? Does the phone need to be on? I tried that too.

You could use the SIM removal tool that came with your iPhone. You push the end of the paper clip straight in and the tray should unlock and partially come out. Doesn't matter if your phone is on or off.