There can be dozens of emotionally charged diminutives (Витенька, Витюня, Витёк) for every name, which can be mind boggling. They are usually quite intimate, and it's safer not to use them if you are not sure if it's acceptable. Later, as you progress you'll be able to guess what diminutive will "fit" a certain name and create your own by transforming "short forms" of names.

"Short forms" of traditional names (Петя, Витя, etc.) are much more "neutral", and do not sound too awkward even when used at the wrong setting. They are widely used and are usually limited to 1-2 for every name. It's entirely possible (and necessary) to learn to recognize most of them. A child, for example, almost always introduces himself by the said "short form", even when speaking to a stranger. So it's very useful to know diminutives for most popular names.

Some examples of neutral diminutives:

Алексей - Алёша,
Сергей - Серёжа
Пётр - Петя
Фёдор - Федя
Вячеслав - Слава (Слава also can be a diminutive for other names with "-слав" in it - Станислав, Владислав, etc.)
Константин - Костя
Иннокентий - Кеша
Борис - Боря
Леонид - Лёня
Степан - Стёпа

Елена - Лена
Ирина - Ира
Наталья - Наташа
Ольга - Оля
Светлана - Света
Татьяна - Таня
Мария - Маша
Екатерина - Катя
Анна - Аня
Елизавета - Лиза

There are some names, that can be used both as informal and as formal (with patronymics added). They still have other ("emotional" or joking) diminutives, of course.
Examples:

Андрей
Игорь
Максим

Марина
Жанна
Тамара