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:
In addition to @MBR's answer; there are other (less confusing) ways to represent the data. This paper describes the method used; basically they plot the position and motion of the stars, or of hydrogen clouds. The hydrogen distribution of our galaxy looks like this:
Now there's a lot of noise in these measurements, but generally you can see that the hydrogen clumps together in spirals.
One additional source of evidence comes from other spiral galaxies. We can see what they look like, how they are composed and we can see that the evidence we have from observations of our own galaxy fits with those galaxies. It's not as strong as the observational evidence shown elsewhere in the answers here, but it does support them.
The clues we have to the shape of the Milky Way are:
1) When you look toward the galactic center with your eye, you see a long, thin strip. This suggests a disk seen edge-on, rather than a ellipsoid or another shape. We can also detect the bulge at the center. Since we see spiral galaxies which are disks with central bulges, this is a bit of a tipoff.
2) When we measure velocities of stars and gas in our galaxy, we see an overall rotational motion greater than random motions. This is another characteristic of a spiral.
3) The gas fraction, color, and dust content of our galaxy are spiral-like.
So, overall, it's a pretty convincing argument. Of course, we have to assume our galaxy is not completely unlike the other galaxies we see--but I suppose once a civilization has accepted that it does not occupy any special place in the universe, arguments about similarity seem sensible.