Potting up: Seedlings can remain in their original containers until you are ready to plant them in their permanent spots. However it is common to move the seedlings into a larger pot once several sets of leaves have formed and the seedling is a couple of inches tall. This is called "potting up" and it allows the roots more room to develop. Three to four inch pots are good sizes to pot up to, allowing plenty of room for root growth.
Thinning: If more than one seedling is growing in the same pot, either separate the seedlings into individual pots or cut off all but the strongest seedling. Don't try to pull out the extra seedlings, since this might hurt the roots of the seedling you are keeping.