When a company increases the number of shares. This is done to increase liquidity or to lower the price on a per share basis, making it more affordable.
With a stock split, the number of outstanding shares will increase by some multiple. The per share price will be reduced by the same, resulting in no change to the market cap.
Typically, the splits are on a common ratio basis, i.e. 2-for-1 or 3-for-1. This means that an indivudal holding 100 shares, under a 2-for-1 split, would end up with 200. There is no net gain, in theory, since the total number of shares outstanding would also double.
A reverse split is the same in principle. The key difference is it reduces the number of shares outstanding, hoping to drive up the per share price.