The most common kind of trust in America today is the revocable living trust: a trust set up by someone who wants to save family members the hassle, delay, and expense of probate after his or her death. In essence, it's a will substitute, and the successor trustee's job is similar to that of the executor of a will.
Simple living trusts are designed solely to avoid probate; they don't affect taxes or long-term property management or anything else. Basically, if you take over as trustee when the person who made the trust (the settlor) dies, your job is to promptly distribute trust property to the people who inherit it. It should take no more than a few months. Here are some of the nuts and bolts of what you'll be doing.
Learn more about serving as a trustee.