egrep is 100% equivalent to grep -E
fgrep is 100% equivalent to grep -F
Historically these switches were provided in separate binaries. On some really old Unix systems you will find that you need to call the separate binaries, but on all modern systems the switches are preferred. The man page for grep has details about this.
As for what they do, -E switches grep into a special mode so that the expression is evaluated as an ERE (Extended Regular Expression) as opposed to its normal pattern matching. Details of this syntax are on the man page.
-E, --extended-regexp
Interpret PATTERN as an extended regular expression
The -F switch switches grep into a different mode where it accepts a pattern to match, but then splits that pattern up into one search string per line and does an OR search on any of the strings without doing any special pattern matching.
-F, --fixed-strings
Interpret PATTERN as a list of fixed strings, separated by newlines, any of which is to be matched.
http://unix.stackexchange.com/questions/17949/what-is-the-difference-between-grep-egrep-and-fgrep
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