Do departments look at MOOC's certificates favorably during the graduate admissions process? Will it give me any advantage over students who don't have them if I am trying to enter a PhD program?
You're free to mention the certificate in your application. How much weight to give to it is presumably up to individual committee members. I think unless the certificate pertains to a specific skill that's in demand for the program, you shouldn't expect to get much of a boost from having one. At the moment, people don't have much experience with such certificates, and will be more inclined to look at indicators they know better, like grades and GREs. It's possible this will change a bit in the future, but it will likely move pretty slowly.
What sort of advantage do you have in mind? MOOCs could help a little in addressing weaknesses in your application: if there's a standard course you were unable to take, then it could be helpful to be able to say you learned the material by other means, and a MOOC certificate might carry a little more weight than completely independent reading. However, on a scale from saying you read a book to getting a strong letter of recommendation, a MOOC certificate is much closer to saying you read a book.
Beyond that, it can't hurt to list MOOC certificates on your CV, but I doubt they'll make any difference. If you're going to graduate school, you should have spent time on many different sorts of learning: lots of formal classes, extensive discussions with peers, independent reading, and ideally working with a faculty member on something (a senior thesis, undergraduate research, etc.). Adding a few online classes is just not a big deal, especially if people are unsure of what the standards are. Compared with things like letters of recommendation or research experience, MOOCs disappear in the noise.
If the MOOC certificate is in a closely related field, it will help. If it is in something like "Underwater Basket Weaving", it may not give you an advantage over anyone else, unless of course you are entering a program in Underwater Basket Weaving. Or something similar, such as Underwater Knitting.