In "The Merchant of Venice", what is so important about the ring given by Portia to Bassanio? What does it symbolize?

Well, when Portia gives Bassanio the ring, she says this:

This house, these servants, and this same myself,
Are yours, my lord,—I give them with this ring;
Which when you part from, lose, or give away,
Let it presage the ruin of your love,
And be my vantage to exclaim on you.

And he replies

                                   But when this ring
Parts from this finger, then parts life from hence;
O, then be bold to say, Bassanio's dead.

You've got the word right: the ring does indeed "symbolize" Bassanio's ownership of Portia's house, her servants and herself (a man "owns" a wife, according to the Christian marriage ceremony). If Bassanio meets with the conditions (i.e. wears the ring) the goods are his - if he breaks them (takes it off) then their "love", and marriage, we assume, is over.

The ring is another "bond". It ties Bassanio to an agreement in return for riches. Just like Antonio's bond made with Shylock. Ominous stuff?

Yes. And it gets worse. To the Elizabethans "ring" referred to the female genitals (re-read Viola's "I left no ring with her..." and tell me it doesn't make more sense now that you know that!). Portia is also, in effect, giving her body to Bassanio as part of their marriage. Portia herself is part of the goods. Now that's chauvinism for you.