Alkaline batteries are the most common batteries you will come across. They are the ones sold in every store, so they're great for projects that need to be 'user serviceable.' They have higher power density than NiCads and slightly better power density than NiMH. However, they are one-time use. Cells are 1.5V, and available in sizes from coin cells to AAAA to D cell. One nice thing about having multiple sizes with a standard voltage is that you can always just specify the next size up when you need more capacity and capability. You'll get the published capacity rate if they are discharged at about 0.1C.
6V lantern batteries (image above) are very large alkalines made of a couple large cells, they're rather convenient in that they're available in many stores, have massive capacity and capability and you can clip/solder onto their tabs pretty easily.
9V batteries are a strange case: they're actually made of 6 very small 1.5V batteries, pretty much the size of coin cells. As a result they have very low capacity and capability and are very expensive. If you are drawing more than 20mA then they are probably not a good idea to use.
Prices: AA size battery costs about $1 and has up to 3000 mAh of charge capacity.
Power density: 100 Wh/kg
Pros: Popular, well known, safe, long shelf life
Cons: Non-rechargeable, low-capability