Hitchcock was known for his suspense films, and yes, the film has a boatload of suspense.
But, The Birds was his first shot at a straight-up horror-genre film. He nailed it: it's chock full of horror-movie tropes. For example, the overwrought villagers who mutter darkly among themselves ("I think you're the cause of all of this. I think you're evil. EVIL!") hark back to at least Bram Stoker's Dracula. And, of course, horror films love monsters—and The Birds has one of the great monsters ever in its hordes of flapping, squawking antagonists.
Hitchcock being Hitchcock, he plays with the horror genre in daring ways:
First, of course, the monsters are just ordinary birds; the cheerfully chirping friends we see every day suddenly turn malevolent and threatening.
Plus, the real horror is that we never learn why the birds are attacking and killing. Usually, by the end of a horror film, you know what Dracula is or where the evil, slimy critter comes from. But, in The Birds, the chaos is never explained—it's just chaos. We're confronted with our total lack of control. And that's horrifying.