As the name implies, "weekly" options are those that expire on Fridays outside of the traditional monthly expiration week (normally the third Friday of each month). Weekly calls and puts are available on a wide variety of popular stocks, which means that every week is an options expiration week when you are trading weeklies.
As a result, you have the opportunity to fine-tune the time frame of your trades to line up almost exactly with your forecast for the stock's expected move -- which means you can be stingy about how much time value you're buying. And since time decay (i.e., the non-linear decline in an option's value as expiration approaches) is the enemy of the option buyer, the ability to exercise tight control over how much time value you're actually purchasing is a major benefit.
Most stocks with listed weekly options have contracts available to trade up to six weeks out -- which means you can trade an expected overnight move using the nearest expiration date, or you can look to take part in a developing uptrend (or downtrend) by buying just over a month's worth of time premium.