In Spanish, the letter 'u', pronounced like the English w, after a g (gey--"hay" in English) before an e ("ay") or i ("ee"). A Spanish 'g' followed by an 'e' or 'i' becomes like the letter jota (j, pronounced "HO-ta"), so gémelo (twin) is pronounced "HAE-mae-loe".
The g can be turned into a hard "g" sound by adding a 'u': guerra (war) is pronounced "GAE-rra" (with that lovely trilled "rr".
A 'u' between a 'g' and another vowel, however, forms a dipthong sounding like a "w-vowel" combination: Uruguay, pronounced "oo-roo-GWAY"º.
Now, what if you want this with an 'e' or 'i'? That is, something sounding like "gwae"? Then you add a dieresis to the 'u', making it ü! Result: pingüino (penguin), pronounced "peen-GWEE-noe".
Oh, and in HTML it's ü (even though it's a dieresis, not an umlaut (Ü for capital Ü)
º For some reason, this doesn't sound right according to general stress rules, but that's how I remember my high school Spanish teachers pronouncing it...