The "original" OP protocol is set forth at http://www.oilpulling.com.
It calls for using either sunflower or sesame oil (I use the Spectrum organic unrefined sesame -- about $6 per bottle -- for OP for the most part...you can spend a lot of money on really nice oils and it is fun to do so, but sesame, for me, is the best/most reliable/best price point), and that you swish/move the oil around in your mouth for 20 minutes once per day before eating. When there are newcomers to the site, I usually suggest they follow that protocol for a couple of weeks to acclimate, and then to try different oils/doing it more often in the day and so forth.
One assumes that in India (OP being an old Ayurvedic practice; you can find referenced to OP in the ancient texts) for a long time they were using non-processed oil. On oilpulling.com it's not really clear. Our standard approach at OP has been to use organic/unrefined when possible, and not to use "industrial" oils like corn/vegetable, soy, canola, peanut. Here is a great site with information on sesame oil, cold-pressed oils, and some references to OP:
Olive was a big favorite of mine for a while (my mom gave me half a gallon a while ago, so I thought I was set for a long time...) until it turned my teeth -- temporarily -- a sort of chartreuse cast. Others have had this experience, so, while we all love olive oil for ingesting and you can get some really great olive oils, it is not a great OP oil. Unless you crave greenish teeth!
Apart from sunflower and sesame, others have used oils such as flax seed, hemp, coconut, walnut, avocado -- well, any kind of nut or seed oil. Coconut and sesame have brightened my teeth the most.
After nearly 2 years of experience with OP, I think the most important therapeutic factor is not type of oil but consistency of practice. I can fall off the beam for a little while, and when I start oil pulling again I am always surpised at how much better I feel again. It's been great for my: sinuses, mood, skin, teeth/mouth/swollen glands (I don't get them anymore, which tells me some sort of low-level infection is gone...). All that said, sesame would still be my "desert-island" oil -- if I had to pick just one. Actually, sesame has about replaced olive oil in our household for cooking, OP, face-cleaning/moisturizing, sunscreen, and so forth. It is a terrific oil!
One thing I do sometimes is to add some oil of oregano to the OP oil I am using. It's a good antibacterial agent.
I hope this clears things up a little. It is a lot easier to do once you start just doing it -- it was easy for me to get bogged down in what kind of oil, how exactly to "swish," etc. -- I think I used black sesame from the Asian food market, which we had on hand -- it was very icky and viscous and in fact tasted sort of burnt, certainly not organic, probably refined to a degree -- but, I remember being amazed by the results within a short time.