A hydrometer is a useful piece of brewing equipment, but is certainly not required to brew beer. In fact, there are some homebrewers who never use a hydrometer when they brew.
But, there are a few reasons why a hydrometer can be useful. You cannot always see when fermentation has stopped, and a hydrometer is the only way to know for sure when the beer is finished fermenting.
The other reason is to check the alcohol by volume (ABV) of your finished beer. If all you really care about is what the beer tastes like and not concerned about the ABV at all, then you can skip the hydrometer.
You should be able to get this hydrometer and testing jar kit for less than $15.00.