Why should I care about these so-called Representatives?

Every wallet seems to have this representative thing. Should I care? How important is it? I don’t give them my nano right?

Please have a look at this video on How to Change a Representative

The representatives are entities of the nano-node application which vote on the outcome of any conflicting transaction by having a certain delegated nano weight assigned to it.

An account’s representative has no access to the account’s funds. The representative only votes on behalf of the account.

More info can be found in the developer resources about Representatives