Retiring an offset means it's claimed against an individual or organization's carbon footprint, and it has been taken out of circulation to ensure it can only be claimed once.
Here's the lifecyle of a carbon offsets:
First, a "project developer" works with a "project owner" to develop a carbon offset project. They must bring in scientists / consultants to verify their plans, and they submit 100+ page project description document to a "registry" (e.g. Verified Carbon Standard). Here is the project description document of the Afognak project we support.
The registry analyzes the project to see if it passes certain criteria (e.g. it's additional, reasonably permanent, etc.) then they give the thumbs up.
The project developer develops the project (e.g. plants the trees, gives the cows a methane abatement supplement, change a landfill to capture the methane, etc.).
Usually a third party comes out to verify everything is legitimate.
Then, a year after commencing, the registry may "issue" carbon offsets. Each represents one ton of CO2e either sequestered or prevented from being emitted in the first place. Each of these offsets has a unique serial number and is tracked in a public database. Once it's issued, it can be bought/sold like any other commodity.
Let's say the project developer is acting like a wholesaler. They sell them to someone like Carbon Neutral Indiana. Then we retail them to the end consumer.
Finally, when any of our clients (a household, a business, etc.) wants to claim those offsets against their carbon footprints, we "retire" the offsets on their behalf. This means they are taken out of circulation and can't be bought / sold by anyone else. Here's an example of what the retirement record looks like.
An analogy for all of this is a coupon for a haircut. You could pass the coupon around, but eventually someone is going to get a haircut. At that point, the coupon is torn up so nobody else can use it. In the case of an offset it isn't torn up by "retired."