Maria (full name) is the same as Masha (short version), or Mashen'ka (pet name);
Lubov' (full) - Luba (short), Lubochka, Lubon'ka (pet names);
Alexandra (full) - Sasha, Alex (short), Sashen'ka, Sanya (pet names);
Ekaterina - Katya (short), Katen'ka, Katusha (pet names);
Elena - Lena (short), Lenochka (pet name);
Anna - Anya (short), Anyuta, Anechka, Annushka (pet names);
Tatyana - Tanya (short), Tanusha, Tanechka (pet names);
Anastasia - Nastya (short), Nasten'ka, Nastyusha (pet names);
Elizaveta - Liza (short), Lizochka, Lizon'ka (pet names) etc.
Russian short names are so short that sometimes they can stand for several full Russian names, and vice versa. For instance, Anya could be an Anna and also an Anastasia. They are used in daily life, within the family and with friends, especially if it is a child or a teenager.
Patronymics play a very important role in the language and culture. If you want to show your respect to a person, to address a superior or just someone older than you, you are to use a combination of the first name and the patronymic. In Russian you never call a person by the surname, like Mr. Smirnov. It is considered to be highly impolite and is only used by teachers to call out their students, or in official situations where the surname serves to identify you, for example, when you wait in a line in front of your doctor's cabinet and the nurse calls out your name.
Of course you can use the first name if you talk to your friend or someone younger. However, with your colleagues you are often required to find out whether you can actually do without the patronymic - it is indeed hard to remember so many! In international communication, the Western standard is used, and Western people are very surprised to find out about patronymics and their use. Some actually make up patronymics for themselves, which sounds very funny - James Edwardovich, for instance.
Many wonder whether it is really necessary to use both the name and the patronymic even for university teachers, at the time when the students are almost adults. However, research shows that in Russia, when you stick to the first name in a situation where you are subordinate, a lot of respect and distance is lost along the way - and respect plus distance are a major feature of Russian education.