How did the 2 dollar bill start and why?

The first $2 bills at the national level were U.S. Notes issued in 1862. Private banks may have issued their own at an earlier date, however.

The $2 denomination makes a lot of sense in a decimal currency system because 2 is a factor of 10, and having denominations that correspond to exact fractions and multiples of the basic unit simplifies change-making. Most countries that use decimal currency have 2-unit bills or coins, such as $2 in Canada, Australia and New Zealand, £2 in Britain, and €2 in the EU. In those countries the denomination circulates widely and regularly.

In the US $2 bills circulated at first, but during the late 19th century were heavily used as payment for horse racing bets and, uh, "sexual services" where $2 was the going rate in those pre-inflation days. Many people came to associate them with crime and refused to spend them so the denomination fell out of favor. Even today some people mistakenly believe that the bills are (pick one) no longer in circulation, used only in crime, or are "cursed", all of which are false.