If you're asked to serve as the trustee of someone's trust, it's an honor--it means the person who created the trust judged you reliable, honest, and capable. But serving as a trustee can entail a lot of responsibility, and you should carefully consider the workload you're taking on before you accept the job.
The good news is that if you're asked to be the "successor trustee" of a simple living trust, you shouldn't have a lot of work. A simple living trust works like a will--it says who should inherit what--but distributing property after the death is easier than with a will. Instead of going through probate court, you can just follow the instructions in the trust document and turn over trust property to the people who inherit it. The whole thing can probably be done in weeks or a few months. Of course, sometimes there are complications, and if there's no will or if you're also named the executor of the will (that's common), then you'll have more work to do.
If the trust is more complicated--for example, it's meant to hold property for children until they grow up--then your job will be more complicated as well. You'll have to make investment decisions, deal with beneficiaries, and file tax returns, possibly for many years.