The galaxy everyone should know best is our Milky Way galaxy. Look up into the sky pretty much on any clear night and you should be able to see a milky band of fuzzy light stretching north-ish to south-ish. It'll look something like this:
You can see fuzzy light from stars too far away to resolve individually, you'll see dark dust clouds obscuring the middle and do you see those two fuzzy patches to the left? Those are the Magellanic Clouds. They were named "clouds" by pre-techological humans, even though they're galaxies in their own right, busy tagging along with the Milky Way.
The word 'galaxy' comes from the name Milky Way in Greek: galaxias (γαλαξίας). Rumour has it a goddess was breastfeeding and the milk from her breasts sprayed across the heavens.
Aside from the Milky Way and the Magellanic Clouds, there is one other galaxy that you can see with the naked eye, the Andromeda Galaxy (M31). Its picture is often used as the poster child for most galaxies:
Granted, it's so far away you can only see the core with the naked eye, but it can be seen here (M31 is the catalogue name for the Andromeda Galaxy) in the Andromeda constellation, just north of the Great Square of Pegasus:
But if it was luminous enough for the whole thing to be seen with the naked eye, it'd actually look this big:
Yeah. That's awesome.
Galaxies come in several shapes: barred spirals (like our Milky Way), spirals (like Andromeda), ellipticals (big egg-shaped clouds), irregulars (like the Magellanic Clouds) and lenticulars (which look like Mexican sombreros).
We're currently riding along in an outside spiral arm of the Milky Way. If you look up at the constellation Orion, you're looking through the spiral arm out into Outer Space. If, six months later, you stare at the constellation Sagittarius, you're looking inward to the center of the galaxy.
There's a supermassive black hole in the middle of our galaxy called Sagittarius A* (A-star). It's a bit hard to see with the naked eye, so you'll just have to rely on those scientists who have access to X-ray observatories and other ways of piercing the thick dust clouds of the galactic core. Here's how we know there's a black hole in the middle of our galaxy--by its gravitational influence on surrounding stars.
Wanna go hardcore over the galactic core? See what Sgr A* is up to.
What's your favourite galaxy? Got any pics?
Her Grace, as a child, was once very disappointed that she couldn't see the Andromeda Galaxy due to a bad spate of light pollution. Alas, this is still an issue for her.