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Variant transliterations arise when transcribing a name from another alphabet into English. I am guessing you saw that in the Slavic names, so let me exlain.
The Cyrillic alphabet contains some letters that have no definite counterparts in English. For example (you need to have your encoding on Cyrillic to read those):

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þ - this one sounds like "you". It can be represented as "u", "iu" or "yu". That is how you can get Ludmila or Lyudmila.

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ÿ - this is the sound when you say "i" as in "sit" and "a" as in "apple" together. It can be represented as "ia" or "ya". So, Veniamin and Venyamin are both variants of the same Cyrillic spelling.

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In addition:
The Cyrillic alphabet doesn't have a letter for the "x" sound, so names like Alexander are spelled with "ks": Aleksander. That is why such names can be written in English either with "ks" or with "x"

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At the end of a word the Cyrillic "è" can be represented with either "i" or "y". That is why you can write Evgeni and Evgeny.

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In Russian the "e" in the beginning of a name is pronounced as the "ye" in "yes". That is why in the Russian names you see Ekaterina and Yekaterina, Evgeni and Yevgeni, etc.

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I hope this clears things a bit :P