The Wikipedia describes a galaxy as "a system of stars, stellar remnants, interstellar gas, dust, and dark matter bound together by gravity. The word is derived from the Greek galaxias, literally 'milky', a reference to the Milky Way galaxy that contains the Solar System."
The Universe consists of billions of Galaxies. A Galaxy is a -often spiral shaped- cluster of (billions of) solar systems. Our local Galaxy is called the Milky Way, and consists of 100 billion (or 100 000 million) solar systems. It is spiral shaped. Our Sun and its planets are located on the outside of the spiral. Our closest galactic neighbor is the Andromeda Galaxy, at a distance of 2.2 million light years.
Many theories state that our universe is just one of many universes, that there would indeed be several universes, often referred to as a multiverse.
Note on semantics: in English, the word galaxy is sometimes also used in the narrower sense of a solar system, or of a cluster of solar systems.