The way Russian and other languages which use the Cyrillic alphabet are transliterated into Latin, the way the English language uses it, generally follows the standard set by the Library of Congress. It is simplified a bit, so it can be typed out using only the ASCII character set (all the characters US-English keyboards can write). This requires using multi-letter groups to replace certain Cyrillic letters.
The scheme is a bit outdated. Since the breakup of the USSR, the West has become more familiar with current Russian usage and pronunciation. To properly represent Russian names for laymen (not linguists), and I want Latinization to best represent how the language is pronounced, not necessarily written, I would make a few changes.
(This does not take into account two features of standard Russian pronunciation: the devoicing of voiced stops and fricatives at the end of words, and the neutralization of unstressed vowels, i.e. why до свидания is pronounced "da svidaniya".)
- Ее: At the beginning of words, after vowels and after Ьь, I would write this as "ye": Достоевский > Dostoyevskii. After consonants (where it usually, but not always, palatizes or softens the preceding consonant), I'd have it simply as 'e': Евгений Онегин > Yevgenii Onegin.
- Ёё: When pronounced as "yo" in "your" (and in that situation, always stressed), the two dots are optional in everyday writing. Still, I would always transliterate this as "yo", and simply 'o' after consonants that are always soft or hard: Пётр > Pyotr; Горбачёв > Gorbachov.
- Жж: The usual practice is to write this as "zh", and I would do the same. I have seen it as 'j', but I feel that would be too easily confused as the 'y' or "dzh" sound. Therefore, I would have, Нижинский > Nizhinskii.
- Йй: This is usually found after vowels, as part of a diphthong, and I would write it as 'i': Николай > Nikolai; Дмитрий > Dmitrii. Before vowels, I would write it as 'y'.
- Хх: Though this is sometimes recorded as 'h' (as in Serbo-Croatian), I would continue to write it as "kh" to avoid confusion: Чехов > Chekhov.
- Цц: This should be continued to be transliterated as "ts", not as "tz", "cz" or 'c' (before front vowels in the latter case): царь > tsar'.
- Щщ: Traditionally, this letter has been pronounced as "shch", and still is in Ukrainian. In current Russian, however, it is pronounced as a long, soft "sh" sound. I would transliterate this not as "shch", but as "sch", still distinct from the always-hard "sh", used for Шш: Хрущёв > Khruschov; Щедрин > Schedrin.
- Ъъ: Since the spelling reforms of 1918, this letter has become exceptionally rare. It has no pronunciation of its own, and only serves to block palatization (softening) of the final consonant of a couple of prefixes. Therefore, I would not transliterate the letter itself, but write the following vowel with a 'y': съезд > syezd.
- Ьь: On the other hand, this letter is frequently used--to palatize (soften) a preceding consonant. I would maintain the current practice of transliterating this as an apostrophe: цар > tsar'. However, after consonants that are either always hard or always soft, I would ignore it: ночь > noch. (Here, the letter serves to indicate that the noun is feminine, despite ending in a consonant.)
- Ээ: This letter is always pronounced 'e' and does not soften a preceding consonant. It is usually, but not always, found at the beginning of words, more often in foreign loans. I would write it simply as 'e': электроника > elektronika; поэт > poet.
