It depends on the type of animal and the organization you register with, I think. My original goats came with ADGA registration, so any of their babies can be registered as well. You basically fill out a form and send in a fee. Each animal has to have a tattoo in their ears identifying it. If you are a member, they have a lot of member services. (Printable forms, registering online, looking up goats by member or ID #, researching lineage online, stuff like that.)
For a large breeder, I think it would be beneficial. As a member in their database, they get more exposure. Through the organization, you can have things done like milk testing, dna testing, earning milk stars for production, (for their records, so you can say your baby comes from a triple star goat, or whatever), and there is a whole slew of shows they put on to get you exposure and titles in the 'goat world'.

For me personally, I don't need all that. I register purely to make it easier to sell babies. Just to have the option available to the buyer.

I think, without the frills, it boils down to having a record of lineage and making sure the animal is purebred.