There are various types of nuclear weapons, and they also vary in terms of size and explosive yield, but the basic structure and function is the same.
In layman's terms, let's start with the "pit", or the plutonium core of the nuclear warhead. This man-made element is one of the most efficient materials we have found for nuclear weapon design. It is highly radioactive and atomically unstable. A typical weapon (yield of about one megaton) has a plutonium core of about 16 pounds. The size of the fissile material is critical, too large or too small and the bomb will not function.
Surrounding this pit is a series of shaped plastic explosive charges, specifically engineered to implode rather than explode. This is one of the things that makes nuclear weapons such complex machines, as the amount of shaped explosive, its design, and the timing and sequence of the implosion are all key in order to make a nuclear weapon function.
The implosion compresses the pit to a mere fraction of its original size, generating immense amounts of heat and pressure, which fractures the shell of the atoms at the center of the pit. The energy that is subsequently released is so great as to split the shells of the remaining plutonium atoms, and thus the main explosive yield of the weapon detonates.
The blast wave is immensely destructive, as is the heat released and the radioactive fallout that follows. The energy is so intense near ground zero that the ground actually melts and fuses.
Variations include Hydrogen bombs (thermonuclear devices) and the inclusion of Tritium as a yield multiplier. Hydrogen bombs use fusion instead of fission, and this requires very, very high temperatures, in the millions of degree Fahrenheit. The only way to create that amount of heat in a man-made reaction is with a fission bomb. So yes, a fusion bomb has a tiny fission bomb at its core.