Is the atomic bomb a nuclear bomb?

Yes, both atomic and nuclear describe what mechanism causes an explosion. It is causes by atoms decaying, or the nucleus being split apart and releasing energy.

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More exactly, a nuclear weapon is one which derives its power from energy released by breaking intra-nuclear bonds (i.e. the energy that holds neutrons and protons together inside an atom's nucleus). There are four types (currently) of nuclear weapons:

An atomic bomb is more properly called a fission weapon, in that it derives power from splitting a heavy element (usually U-238 or P-239) into smaller elements.

A thermonuclear bomb is one that relies on fusion, where lightweight elements (isotopes of Hydrogen) are pushed together. Currently, all such weapons require a small atomic bomb to act as the trigger for the fusion reaction, but the vast majority of their yield is from fusion.

A hybrid weapon, also called a fission-fusion-fission weapon, is a thermonuclear bomb wrapped in a uranium shell to boost the yield by using escaping neutrons from the fusion reaction to ignite the uranium shell's fission.

A boosted atomic weapon is a normal atomic bomb which has gaseous tritium (a hydrogen isotope) injected just prior to detonation. This gas undergoes fusion, increasing the yield of the weapon.