It's difficult for native English-speakers too! It's quite an unusual word, rarely used in speech except perhaps for comic effect. It means, or less, "use", both as a verb and a noun, but its usage is different.
As a noun, its most common use is in the phrase "to no avail" which means the same as "in vain" or "to no effect": "I tried using the key to open the door, but to no avail." You could equally well say "... but it didn't work."
As a verb, the construction "to avail oneself of" means the same as "to use" or "to take advantage of". Thus, "I will avail myself of the facilities in this hotel." means that you'll be drinking in the bar, eating in the restaurant, getting your hair done, taking showers and so on.