They are called lookarounds; they allow you to assert if a pattern matches or not, without actually making the match. There are 4 basic lookarounds:
- Positive lookarounds: see if we CAN match the pattern...
- (?=pattern)- ... to the right of current position (look ahead)
- (?<=pattern)- ... to the left of current position (look behind)
 
- Negative lookarounds - see if we can NOT match the pattern
- (?!pattern)- ... to the right
- (?<!pattern)- ... to the left
 
As an easy reminder, for a lookaround:
- =is positive,- !is negative
- <is look behind, otherwise it's look ahead
References
But why use lookarounds?
One might argue that lookarounds in the pattern above aren't necessary, and #([^#]+)# will do the job just fine (extracting the string captured by \1 to get the non-#).
Not quite. The difference is that since a lookaround doesn't match the #, it can be "used" again by the next attempt to find a match. Simplistically speaking, lookarounds allow "matches" to overlap.
Consider the following input string:
and #one# and #two# and #three#four#
Now, #([a-z]+)# will give the following matches (as seen on rubular.com):
and #one# and #two# and #three#four#
    \___/     \___/     \_____/
Compare this with (?<=#)[a-z]+(?=#), which matches:
and #one# and #two# and #three#four#
     \_/       \_/       \___/ \__/
Unfortunately this can't be demonstrated on rubular.com, since it doesn't support lookbehind. However, it does support lookahead, so we can do something similar with #([a-z]+)(?=#), which matches (as seen on rubular.com):
and #one# and #two# and #three#four#
    \__/      \__/      \____/\___/
References