Chlorination of Water
Passing chlorine gas through water yields a solution of $\ce{Cl2}$, hydrochloric acid ($\ce{HCl}$) and hypochlorous acid ($\ce{HOCl}$). The latter two are formed by reaction of chlorine with water under disproportionation.

$$\ce{Cl2 + H2O \rightleftharpoons HCl + HOCl}$$

Bleaches are chemicals which remove color by oxidation of dyes. $\ce{Cl2}$ and $\ce{HOCl}$ themselves are already strong oxidizers, but adding alkali metal hydroxides to an aqueous chlorine solution increases the solubility of $\ce{Cl2}$, and chlorine bleach is obtained, which contains alkali metal hypochlorite as active ingredient. 

$$\ce{Cl2 + OH- \rightleftharpoons ClO- + Cl- + H2O}$$

Sodium hypochlorite is produced by this reaction and common household bleach is an aqueous 3-8 mass-% solution of the salt. The necessary chlorine gas is actually produced by electrolysis of $\ce{NaCl}$. 

While gaseous $\ce{Cl2}$ itself can act as a bleaching agent by oxidation, the use of hypochlorite solutions has the benefit of avoiding handling of the poisonous gas. $\ce{H2}$ is a much better reducing agent, and therefore unlikely to be useful for bleaching.

A mixture of $\ce{H2}$ and $\ce{Cl2}$ will react to form gaseous hydrogen chloride when exposed to UV light. Hydrochloric acid is obtained when gaseous $\ce{HCl}$ is dissolved in water.

$$\ce{HCl + H2O \rightleftharpoons H3O+ + Cl-}$$