There are many ways to avoid probate, and you'd be best served by reading up on the topic of estate planning and then consulting a professional. In brief, though, one way to avoid probate is through a living trust or a life estate trust. With a trust, you're actually formally transferring the title of various properties to your heirs before you die. You retain control over it while you're alive, but it technically belongs to the trust. Once you die, a trustee passes it on to your specified heirs. This tends to be a fairly quick and simple process.
Also, many states permit a certain amount of property to be inherited without going through probate. In some states the limit is $100,000, for example.
Other possibilities include "payable on death" designations on bank accounts (where the contents of the account pass immediately to the designated beneficiary on your death) and retirement accounts such as IRAs and 401(k)s (where you also specify beneficiaries). Thanks to the Uniform Transfer-on-Death Securities Registration Act, most states now permit securities held by the likes of brokerages to pass to beneficiaries without going through probate.
Learn more about estate planning at the (thorough but not entirely objective) estateplanninglinks.com website and on our Estate Planning discussion board.
These articles should also be of interest: