In all this (infuriating) hullaballoo over ebola, I've seen precious little explanation of this. So, let's go! (please god no mainpage, I'm not an expert on this by any means)

Biosafety levels (BSL from here on out) 1 through 4 are basically a way to categorize infectious agents on how scary they are and how much risk they pose to the person handling the agent. They are also guidelines on safety measures to take when handling these agents.

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BSL-1: well-characterized agents not known to cause disease in healthy adult humans, and minimal potential hazard to lab workers and the environment.
Examples: canine hepatitis, non-pathogenic E. coli, other non-infectious bacteria
Precautions: gloves and maybe some facial protection. Lab is not separated from the rest of the building. Work done on open benchtop. Nothing exciting.

BSL-2: Agents that cause only mild disease in healthy adults, and/or difficult to contract via aerosol in a lab setting.
Examples: C. difficule, Lyme disease, hepatitis A, B, and C, MRSA
Precautions: Specific training in handling infectious agents and supervision by scientists with advanced training. Limited access to lab when work is happening. Extreme precaution with contaminated sharp objects. If aerosols or splashes are likely, work is done under a hood or in a biological safety cabinet (enclosed, ventilated lab space).

BSL-3: Now we're getting into Serious Business. Agents which may cause serious or fatal disease with inhalation. Serious diseases for which treatments exist.
Examples: Plague, tuberculosis, rabies, West Nile virus
Precautions: Work done under specially designed hoods or biological safety cabinets, PPE (personal protective equipment) is worn, double-door access zones to the lab (although apparently not all BSL-3 facilities have this). Strict ventilation standards. Lab personnel have a lot of specific training and are supervised by experienced scientists.

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BSL-4: The good stuff. Agents that pose a high risk of aerosolized infection, agents which cause severe to fatal disease in humans for which vaccines or treatments are not available.
Examples: ebolavirus, other hemorrhagic fevers, smallpox is considered dangerous enough even though there's a vaccine.
Precautions: Extensive. Positive pressure personnel suit with an external air supply. The lab is shower-in/shower-out to enter/exit. There are multiple airlocks. Major decontamination procedures for air and water going out of the lab. Facility is in a separate building or very separated designated area within a building. Significant amounts of training required. There are only 15 BSL-4 facilities in the US, some of which are still under construction. Three of them are in Texas.

For a fun fact, there's a bunch of researchers and staff at those 15 facilities who handle all these fun agents all day for a living, and they travel and go on cruises and run around among the general public! Just don't tell that to the general public. I've actually met some of them, and nobody got ebola.

And now, editorial time. All these precautions are for working with samples in a lab, not a projectile-vomiting human. You really and truly have to be trained, know what you're doing, and have a healthy fear of the agents to work with them, whether in a test tube or coming out both ends of a person. So figuring just any hospital can step in and handle this is a huge mistake and massive hubris.

However, there's many places that have zero BSL-4 facilities. And no one with training, and no one to train. And those are the places where these diseases spread like wildfire. These places need our help. The solution there is not just "you all should have transferred the patient to a properly equipped hospital as soon as you suspected ebola!" In those places, it's hardly even possible to set up proper isolation wards with airlocks and have people wear suits with their own air supply. They do the best they can with what they've got. And that's where the actual crisis is.