According to my good friend Wiki, symphonic metal is regular metal that incorporates aspects of more traditional music like classical. While they do use electric guitars and can be heavy, they also rely on keyboards. They might even use string instruments or a full orchestra. Also, they often use a strong female vocalist and sometimes choirs.
Also, a lot of the bands seem to come from the Baltic region, although there are other ones. It seems that they often draw from fantasy tales or myths when creating their lyrics. I love the melancholy quality combined with the guitars.
Night Wish was the first band that I came across in this genre. I don't remember how I stumbled across them, but I've been enjoying their stuff ever since. That's led me to look for other bands, and now there's no turning back!
Here's some examples for you to check out.
Bless the Child by Nightwish. It also has a very interesting music video.
Last of the Wilds by Nightwish. If you didn't like the last one, you might like this one. It's a nice instrumental one with an awesome violin part. And just for your amusement, I got a video that has lots of clips of Jack Sparrow!
The Dream by Leaves Eyes. This one even has Spanish subtitles. :)
The Howling by Within Temptation. FYI, this might be a bit sad/creepy for some of you. She also has a very strange spiky collar thing. Still, I like the music.