The Tesla coil is composed of two circuits, the primary and the secondary.
The primary coil is the one at the base of the "tower" of the Tesla coil, and it's usually flat or an inverted cone shape. The primary coil is powered by a high voltage transformer that steps up the voltage from the typical 110 V mains to something like 10-30 kV. The transformer doesn't change the AC frequency. This high voltage is used to charge up a capacitor that is charged and discharged every half a cycle. During discharge, which is usually triggered by a spark gap (so it only discharges when the capacitor is fully charged), the capacitor and the primary coil act as a RLC circuit, so it oscillates at a very high frequency. This is the frequency the Tesla coil will operate at. (The coupling with the secondary coil also interfere in this frequency. It gets pretty complicated!)
The secondary circuit is the big towering thing. At the top of the secondary coil there's usually a large conductor with a large curvature, like a sphere or a torus (donut shape). This acts as a capacitor plate (the other plate being the ground, in a very real sense).
The primary and secondary circuits do not touch. They are coupled only via electromagnetic fields: the primary coil creates an oscillating field that induces a voltage/current on the secondary coil, which is inside the primary coil.
The whole thing is just like pushing a swing: the primary coil does the pushing, and the secondary is the swing. The point of the Tesla coil is to get electrical energy in the primary and "push it" to the secondary, which will oscillate with some energy at a particular frequency until you give it another push.
Ideally, the secondary still has energy in it at the time of the next cycle. So you keep pumping energy to the secondary until there's enough voltage to ionize air and cause sparks. At this point you can't really add more energy as it just forces the secondary to discharge in whatever ways it can.
But to do all of this, just like in a swing, you need to push the right way at the right time. This is the main design problem with a Tesla coil. We call this "tuning" it.
Sometimes, especially if the Tesla coil was poorly built and/or not properly tuned (each Tesla coil should operate at a very particular frequency depending on the characteristics of its components), an electric arc may form between primary and secondary coils.
The AC voltage induced in the secondary coil should peak in magnitude at the very top of the coil, the business end. The bottom of the coil is usually grounded so it is taken as zero potential. If the Tesla coil is not properly tuned, this peak may occur in the middle of the secondary coil, or near the bottom where the primary coil is. This means you'll have a large potential difference between this and the primary coil, which may be enough to ionize air and create a spark between primary and secondary. This is bad!
When this happens, both circuits are not coupled only electromagnetically anymore. Now there's a direct path of current between primary and secondary circuits. What's worse, once you have a spark going that air is ionized, so it's highly conductive, which means current going through the arc tends to sustain it over multiple cycles.
This is pretty bad because it can damage the Tesla coil and much more. The total energy stored in the secondary coil can be hundreds or thousands of times the energy of one cycle of the secondary, and this gets pushed back to the capacitor, the high voltage transformer and even your house's main circuit. This can trigger circuit breakers and even damage other appliances connected to the same immediate electric grid.
It's also bad because it's extra dangerous for you. Now your secondary circuit is also under the direct effect of a 60 Hz, 10-30 kV voltage. This is similar to sticking your hands straight into the transformer, which can output a lot more energy now since you are grounded and pose little resistance to that kind of voltage. There's some complicated issues with how the capacitor behaves here as well, which I'm not fully aware of the details. This stuff can get really complicated to analyze formally.
The main point is that when the primary and secondary are connected, you are exposed to a direct current path straight from the wall socket that can provide enough sustained energy to damage or even kill you. The rest of the circuit also acts in complicated ways to make the situation worse.