Comets grow tails because the Sun’s heat turns their ices into gas, and sunlight pressure and the solar wind push that gas and dust away from the Sun into long, bright streams.
A comet’s solid core, the nucleus, is often only a few miles across and is made of icy chunks and frozen gases with bits of dust embedded in them. Far from the Sun, it stays frozen and has no tail. As the comet moves closer to the Sun, it warms up, its ices change into gas, and it develops a fuzzy atmosphere called a coma, which can extend hundreds of thousands of kilometers. The pressure of sunlight and high-speed solar particles then blows material from the coma away from the Sun. Most comets actually have two tails: a dust tail and an ion, or gas, tail, which is why a bright comet like Hale-Bopp can show two separate streaks.
Source: NASA Science, Comets: Facts
Image: Comet Hale-Bopp, E. Kolmhofer and H. Raab, Johannes-Kepler-Observatory, Linz, CC BY-SA 3.0, via Wikimedia Commons.