Here are some of the ways in which I have started seeds inside...

1. Buy the peat pellet greenhouse with dome.  This is a bit expensive if you are starting alot of seeds, but very easy to use.
2. There are prefilled seed starting trays you can purchase as well, also a bit expenisive if you are doing alot.
3. I recycle things from our office... Some of us use the containers that our pap tests come in... great for starting 25 seeds at a time... also has a cover for the greenhouse effect.
4. Another thing I save from the office is clear plastic bags.  I start seeds in ice cube trays that I find at yard sales and the swap shop.  Just fill the seed with your starting medium, add seeds, water, place inside plastic bags.
5. Small plastic pots can also be used inside plastic bags.
6. Yougurt or other small plastic containers.

Almost anything can be used to start your seeds inside... just make sure your not going to water the windowsill when you water your seedlings!