Like how it is common to see their/there and "could/would/should of" instead of "could/would/should have" in English discussions.
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Due to the way kanji are typed (i.e. using an IME which presents you with candidates from a dictionary), and the fact that Japanese kana usage is by-and-large phonemic (i.e. you write it how you say it), there aren't really many mistakes that are entirely analogous to your/you're or there/their/they're, etc.
Probably the closest thing is typing something into an IME, then inadvertently selecting the incorrect candidate from the list (like say accidentally choosing 以外｛いがい｝ instead of 意外｛いがい｝). This is what is commonly referred to as a 変換ミス｛へんかんミス｝, or "conversion mistake".
Somewhat of a tangent to that phenomenon is the 打ち｛うち｝間違い｛まちがい｝ページ ("Typo Page") that you will can find on Japanese websites (I'm not aware of such a thing for English pages, at least not that's common). These exist primarily to catch poor candidates produced by IMEs or people who have just completely misremembered the -- often stylized -- written form of the name of the website/company they're looking for. You can see a lot of such pages by searching google for 打ち間違いページ.
Addition: ssb also points out above a good tangent of people mislearning/misremembering readings of kanji words, and then not being able to get the IME to produce appropriate candidates as a result (e.g. ふいんき for 雰囲気｛ふんいき｝). This is, however, relatively uncommon in practice because most people will note that something's up when their IME doesn't produce the result they're looking for, or because the IME reinterprets and then effectively hides the mistake by producing the intended kanji; Google IME, for example, produces 雰囲気 when typing ふいんき (with a note about correct usage on the side).