Generally speaking, all Android phones uses a removable Micro SD card where pictures are saved to. The SD card is typically in the same location where the battery is and you must remove the battery 1st in order to remove the SD card. Consult your manual for more clarification if necessary.
With the micro SD card in hand you can transfer the pictures to a computer/laptop with the following two methods:
1. If your PC has a SD card reader, then you need to use a micro SD to SD card adapter. Simply insert the micro SD card into the adapter, then insert the adapter card into the computer's SD slot.
Here's a link if you need to purchase a SD adapter card:
2. With the assistance of a friend's Android phone, simply turn off his/her phone. Then replace his/her micro SD card with yours. Turn on the Android phone and connect it to the computer. Choose the option to use the Android phone as a "disk drive" or "storage device" so that you can access the SD card as an external hard drive.
The pictures should be in the "dcim" directory. There may be two folders called "100MEDIA" and "Camera". In any case, all your pictures should be in the "dcim" directory. Simply select all and copy them to your computer.
Is there anyway to transfer the files that are actually stored on the internal memory? My Droid did use a SD card but I never put one in, so all of my files are on the internal memory. The screen is broken to the point that nothing works and the power button will not work. When I plug it into the computer, all I see is the batter charging screen pops up then goes to sleep, I cannot do anything with the phone.