How can we tell that the Milky Way is a spiral galaxy? From Earth, we can distinguish the type of another galaxy very easily simply by observing the shape, colour, and structure of the galaxy when we image it. But since we are located inside the Milky Way, we cannot get an "outside view". Thus, how can we tell if we are located in either a spiral or an elliptical galaxy?
There are several clues.
The Milky Way is a flat disk
The first one, and the simplest one, is that we live in a disk. As you can see on images like the ones from the 2MASS survey in the infrared range:

We clearly see that we are inside a disk, since what we see when we look around us is a disk seen edge-on. It does not look like an ellipsoid or any other shape; it looks like a disk.
The Milky Way has arms
The other strong clue we have is the velocity of stars of the Milky Way, we can measure. First, we see an overall rotational motions, and, when you look closely to the patterns (like in the following image, that is observations in the radio range of the Milky Way at Green Bank Observatory), you can deduce that stars are even ordered into spirals: