Do you have a table at home or do you plan on going to the pool hall?
Do you have good lighting at the pool hall or home and what's the condition of the cloth?
Is money a factor or is common sense more of a factor?
At home or the pool room the phone/ipod route is a good cheap way to go. The apple products take great video but know that apple saves files in AVCHD and not AVI this could be an issue for depending. Next is they save the files in 100% raw format so expect a 15 minutes session to be up to about 1 gig.
You can use other manufactures in that you can now save the data to a sim card of larger space, keep in mind you will need card readers to view on you computer, again I don't know what your plan is.
Phones are going to have some issues when comes to focusing and capturing light through the lens and crap like that, that's why lighting or table cloth becomes an issue (i.e. if you've ever watched a stream from joe blow and see a bright glare on the table) and here is a nice little tripod that can be use at the pool hall or at home sometimes depending on layout, they are also able to attach to tripod for more hight. (
http://www.ebay.com/sch/i.html?_odk...Xphone+tripod.TRS0&_nkw=phone+tripod&_sacat=0)
If you are going to be doing this at home I say use your laptop and a webcam, I suggest the logitech C920 or so series. (here is what it looks like
https://www.youtube.com/watch?v=q5wAZRiwZm0) that way you save it straight to the hard drive, view it right there and maybe even on the TV via HDMI.
Now again, I don't know you or what you plan is or what you're hoping for or expecting or what you knowledge about this stuff is, soooo.
Lastly you can go the camcorder route, this could be hard or easy. You could over think it and spend the next 3 months looking for one or buying the most expensive one thinking its the best. Again, don't know you so don't know how you're going to approach this.
This is the camera I currently use (
http://www.amazon.com/Canon-VIXIA-Image-Stabilized-Camcorder/dp/B006UMMPGE) in my world this was a win cause I was bought with the intention of capturing my daughters taekwondo, her singing or plays in school and family vacays, oh yea and some pool. I use this tripod (
http://www.ebay.com/itm/Bower-Profe...885?pt=LH_DefaultDomain_0&hash=item4adc89ee65) with it. I tape some of my friends games burn to dvd and give to them from time to time. I shoot everything in 1080 but my practice in MP4. The cam records mp4 for 30 mins a pop so it works great as i spend about that much time on a particular exercise. I use a 32gig card and that get's me 3 hours easy of 1080p but 1080p is not small so expect the same size files as using Apple or i guess shooting in HD anyway.
Don't want to be bother with all the above, just want to record yourself and view it on your own time as easy as pie. Go get a tripod, make a Ustream account, buy the device listed below (wifi connection needed, a good wifi signal) and shoot to your hearts delight and have Ustream save all your data then watch it on the computer when you have time.
http://www.amazon.com/Logitech-Broadcaster-Streaming-Calling-Recording/dp/B009C98NQS