There are hundreds, if not thousands of types of hydrangea, counting all species, subspecies, and cultivars. But in general, you will find 5 or 6 basic types (though there are many more in trade if you know where to look);

Hydrangea macrophylla and Hydrangea serrata are probably what most people think of if you say the word "hydrangea"- these are the large, rounded 'mophead' and 'lacecap' varieties- varying in color, but most often blue, pink, or white.

Mopheads and lacecaps:

Hydrangea arborescens are native to the Americas, with 'Annabelle' being one of the most common cultivars. These shrubs are usually leggy by nature and flower white (either as large mopheads or smaller lacecap forms), or with small heads of fertile sepals. There are also a few pink-flowering or pink-tinged varieties.

Hydrangea arborescens 'Annabelle':

Hydrangea quercifolia or 'Oakleaf' hydrangeas are large, hardy shrubs native to the Americas with conical, white blooms that often age to pink with time. These plants are generally stouter, and slightly more sun-tolerant than macrophyllas and serratas.

Hydrangea quercifolia 'Snowflake':

Hydrangea paniculata, panicled, or 'peegee' hydrangeas are arguably the toughest and largest hydrangeas. These shrubs can get as much as 15'+ in height, and are sometimes called 'tree' hydrangeas (because they can be pruned into a tree form). They are tolerant of full sun even in hot climates, and have tapered, lacy blooms generally comprised of white flowers, some of which age to pink. 'Limelight', and 'Pinky Winky' are 2 relatively recent and popular releases of this species.

Hydrangea paniculata 'Brussel's Lace':

Other moderately common types of hydrangea include climbing hydrangea (Hyd. anomala, Hyd. petiolaris, Hyd. integrifolium, Schizophragma hydrangeoides); Hydrangea aspera (fuzzy-leaved hydrangea), and Hydrangea involucrata (late-blooming fuzzy-leaved hydrangea).

Anyway, I probably went a bit overboard there :) ...but I tend to get overenthusiastic about hydrangeas. I hope you find as much joy and wonder in them as I have- and I'm glad to have another new convert around!

^ (PLANT NERD)
I just realized I listed 'Hyd. integrifolium' as a type of climbing hydrangea. That should read 'Schizophragma integrifolium'.

You were not too far off, wild belief......there is an evergreen climbing hydrangea with a very similar name, Hydrangea integrifolia. Only hardy in very mild zones (8 and above) but in my area, it is far more common than Schizophragma integrifolium, which I have never seen for sale.

Oh! Good to know! The only evergreen climber I've ever seen in person is seemanii, which has been pretty useless for us. You should see the Schiz. integrifolium we've got growing on a fallen tree at the nursery; it's absolutely enormous- I'll snap a photo on Wednesday when I'm back down there.

Completely forgot I said I'd post photos... don't know that anyone will care much, but just in case:

Schizophragma integrifolium:

And again: