Relatively safe and stable. That is, for a country in the developing world with a high rate of poverty, crime and HIV infection, it at least does not have widespread violence, civil war or major political unrest.
As you are a young person, of course it is best if you were traveling there with an established group that is experienced in operating there. It is also often advisable to go to a doctor and get shots to protect you from some of the more infectious diseases you might be exposed to.
Not meaning to discourage you from going, but as a foreigner, you should take normal precautions for your own health and safety.
One way to determine this is to look at the US State Department's travel warnings. The State Department issues travel warnings when they think that some country or some region in a country is not safe for Americans. As their website (http://travel.state.gov/travel/cis_pa_tw/tw/tw_1764.html) tells us, they issue these warnings
when long-term, protracted conditions that make a country dangerous or unstable lead the State Department to recommend that Americans avoid or consider the risk of travel to that country.
Right now, there is no such warning for Tanzania. That would imply that Tanzania is not unsafe or unstable.
Whether or not Tanzania is safe and stable right now depends upon your definition of "safe" and "stable". The government of the country of Tanzania has followed the same structure since 1964, with Jakaya Kikwete as president since 2005. Tanzania is a very poor country from an economic standpoint, with per capita income estimated at $1500 per year. There are many refugees currently living in Tanzania after fleeing Burundi or the Democratic Republic of Congo. Crime is an issue and travelers need to be alert and take precautions, as would be appropriate for travelers entering a foreign culture in most parts of the world. Health precautions would need to start with vaccinations prior to traveling to Tanzania and would continue with use of caution regarding food, drink, and cleanliness of person and facilities while traveling. Again, this advice would be common sense for travel to any third world country, not just Tanzania. With proper preparations and awareness while you are there, the sights and experiences that could be gained from a trip to Tanzania could make for a once-in-a-lifetime adventure!