wildcard - a symbol (usually star/asterisk) which can stand for any letter, so for example typing mums* would come up with mumsnet, mumsnot, mumsabigfatgiraffe etc.
Boolean properties are AND/OR/NOT operators. So
beans AND toast
searches for any posts which both the words "beans" and "toast" appear together
beans OR heinz
searches for any posts with "beans" in, and also any posts with "heinz" in.
beans NOT heinz
will bring you any post which talks about "beans" but won't show any results with the word "heinz" in.
You can also combine them, for example:
beans OR jam AND toast
will bring you posts which all contain the word "toast" but will have some with "beans" and some with "jam".
Phrase searching is just including a phrase in quotes. So if you wanted to search for some hilarious thread which you only remembered a phrase from, like "a cutted up pear" it might come in useful. Especially if you wished to add another phrase from the same thread such as "utter arse of a woman" Since "pear" is going to return you millions of pointless weaning threads, likewise "arse" and "woman" are too broad as search terms.
You can try these out on google, though it complains about the AND operator and also wildcards because it does it automatically. It also uses a minus symbol instead of NOT. The phrase search works well though...