The company usually gives you more shares of stock for every share you own. These are often referred to as 2-for-1 or 3-for-1 stock splits. On rare occasions, a company might have a reverse split and give you fewer shares for each share that you own (Ex: a 1-for-2 stock split).
If the stock is split 2-for-1, you now own 2 shares for every share you own. The total basis of the stock doesn't change. Instead, the value of each share changes.
Ex: Lisa owns 100 shares of a company with a basis of $10 per share. Her total basis for her stock is $1,000 ($10 x 100 shares). After a 2-for-1 stock split, her total basis is still $1,000. However, since she now has 200 shares of stock, each share has a basis of $5 ($1,000 / 200 shares).
A reverse split works the opposite way. A 1-for-2 split reduces the 100 shares you own to 50 shares. Now, each share has a basis of $20 instead of $10, but the total basis is still $1,000 ($20 x 50 shares).