Let's take a quick look at Wikipedia:
For example, biologists may count the number of tree species in a forest: events would be tree observations, exposure would be unit area, and rate would be the number of species per unit area. Demographers may model death rates in geographic areas as the count of deaths divided by person−years. More generally, event rates can be calculated as events per unit time, which allows the observation window to vary for each unit. In these examples, exposure is respectively unit area, person−years and unit time. In Poisson regression this is handled as an offset,
Exposure is a measure on how you want to divide your counts to. Do you want to divide by unit area? volume size? It has nothing to do with Poisson regression. It's something you want to do with your data.
Offset is a modelling technique in Poisson regression. If you don't want to use Poisson regression, you won't have an offset in your model. It's a simple trick in Poisson regression that allows you model for rates without a new statistical framework.
We use offset with the Poisson regression model to adjust for counts of events over time periods, areas and volumes. Details on what exactly offset is mathematically, goto:
When to use an offset in a Poisson regression?
Note how the offset goes to the right side of the equation. The offset is the log of exposure (because we're using the log link).