When All-grain brewing, a hydrometer is vital; it tells you that you have extracted enough sugars from the grain, that you are not extracting tannins, when your primary fermentation is done, and how much alcohol is in the beer.
With Extract brewing, if you follow the recipe you will always have enough sugar in the wort and will know the original gravity. If your beer finishes fermenting you will be able to work out the alcohol content.
BUT The only way you will know if the beer has finished fermenting or sometimes even started is to check the gravity with a Hydrometer. Just because the airlock no longer bubbles does not mean the the fermentation is always finished and complete.
Many people do brew without a Hydrometer but it is not a best practice.