Priceline is basically a bidding site. Nell is right, you do get some good bargains there, it's a good site. But the whole idea is based on not knowing the hotel you bid for until you've given over your c/c details. If you do use it Laila, make sure the hotels you bid on is in the G1 or G2 area, that's central Glasgow.
Laterooms.com do serviced apartments in Glasgow central. Have a google of "serviced apartments Glasgow" and see what comes up.
If you're going on your own, please have a read of the "keeping yourself safe" section on the main website. Especially as your preference is for a serviced apartment.
Good luck and don't hesitate to ask if you have any more questions.