The essence of the game is that you are moving from one to four pawns around the country, landing in cities, and scoring "votes" in the appropriate state which are proportional to the size of the city. Each turn, you move and add votes to the states you land in.
There are many complications that can occur. You can sit out a turn and collect money. With money, you can buy "organization" in a state, acting as a multiplier to all subsequent votes you get in that state. You can buy "advertising" - most useful in buying votes in piddly states that you're too busy to visit. You can buy "surrogates" which are additional pawns who may travel, garnering votes, just like your candidate. You can buy "special flights" that help you get to places without the luck of the dice.