I am working on an arduino project that uses transistor switches to power on high power leds that are powered by a separate 24 volt power supply. My issue is that I have two different grounds, making it difficult to bias the transistors. Can I just connect the arduino ground pin to my external power supply's ground or will this damage the board? How can I end up with just one ground? Thanks.
You should be able to connect your arduino ground to the 24V ground, unless the power supplies are somehow referenced to one another. If one or the other or both of your supplies happens to be a battery, then you don't have a concern. If both supplies are transformer isolated AC supplies, you shouldn't have a problem. Even if both supplies tie their DC return side to the earth-ground lug of the AC mains, you should be ok.
If you bridge the two grounds through a 10K resistor, and a DVM across that resistor shows anything more than a volt or so, your supplies probably aren't isolated enough to just connect the grounds.
If the supplies are electrically isolated then the grounds can be connected. If you are not sure you have isolated supplies I would not connect them.
Another option is to use optical isolators and keep the grounds separate. This can be quite a few additional parts but is a very safe way to interface circuits.