The microscope was never really "invented." It's probably more accurate to say that it evolved. And that seems right for what is possibly the most useful and important laboratory tool ever created. The history of the microscope is directly tied into the history of the lens. But it's not completely clear where the first lens came to fruition. Some say the earliest known example is the Nimrud Lens (or the Layard Lens) -- a carved piece of convex crystal that dates back to 750 B.C., Assyria, which historians believe could have been used as a crude magnifying glass (though according to The British Museum, which houses the lens, its convex shape was more likely an accident and the lens is actually just a piece of jewelry). Others say the first lenses weren't created until the 11th century for the purpose of magnifying small text (at the time they were called "reading lenses"). But other historians say the 11th century is late in the lifespan of the lens -- that's because there are references to a "burning-glass" in literature from ancient Greece. One thing we do know for sure is that the first real and true use of eyeglasses can attributed to the Italians around 1260. But before that, well, let's just agree that humans have been using lenses to magnify things for a really, really, really long time.
Given our long history with using lenses to magnify objects and manipulate light, it's kind of surprising that it wasn't until 1590 that a Dutch eyeglass maker and his dad made the world's first actual microscope. Zacharias Janssen and his father Hans made the first sort-of microscope in Middleburg, Holland (Germany and Holland were well-known at the time for being the world's top lens-makers). It looked a bit like a kaleidoscope and consisted of three tubes -- two with a lens on one end and one that was open on either side to hold them together. The lens you looked through was biconvex, meaning it was curved on both sides. The lens that faced the object you wanted to magnify was plano-convex, meaning it was flat on one side and curved on the other. By elongating or shortening the tube, the microscope's user could magnify an image from three to nine times its original size. Kind of like a telescope.
It should be noted that some say Janssen and his father actually didn't invent the first microscope but, rather, their competitor Hans Lippershey, an eyeglass maker who lived nearby actually beat them to the punch. Because even in the 1500s new inventions had intellectual property controversies! Either way, this early device almost immediately set off a flurry of innovations and upgrades.