A second party developer is a company that, while it can make any
video game it pleases, makes them for just one parent company. The parent company usually makes a substantial investment within the second party for the rights to publish the titles exclusively for their console.
A good example of this would be with
Nintendo (which specializes in this type of agreement) and
RareWare. Nintendo bought shares in Rare some years back and now has exclusivity rights to their upcoming games.
On a side note,
Sony recently invested $124M in Squaresoft. Does this make them a second party developer? Not necessarily:
Squaresoft may have signed an agreement with Sony, but as far as anyone knows, they are not exclusive to them.
For information on
first party developers and
third party developers, click the links.