Some devices, like the Motorola Xoom, the Sony Ericsson Xperia Active and the Galaxy Nexus, have an air pressure sensor built into them that measures the barometric air pressure. I have the Barometer HD app installed on my Xoom which seems to give roughly the same reading as my Suunto electronic barometer.
Support for a barometer pressure (and other environmental) sensor was added in Android 2.3 To follow the trend of linking to the related developer resource, I believe this is a TYPE_PRESSURE sensor.
See also this related StackOverflow question Android devices with pressure sensor, and this Gizmodo article for some of the possible uses for the barometer, which includes helping the device to calculate your altitude in GPS apps.
Android Engineer Dan Morrill made a post to Google+ explaining why the Galaxy Nexus had a barometer internally:
I've seen a lot of hay made about the barometer in Galaxy Nexus.
Here's the skinny; it's not really as dramatic or weird as people
The primary purpose of the barometer is (at least, I've been told) to
make GPS lockons faster. Locking on to a GPS involves numerically
solving a 4-dimensional set of linear equations -- 3 dimensions in
space, and time. (Yes, you get accurate time for free if you lock on
to GPS.) Because of the way GPS works, this can take a few minutes.
This goes much faster if you already have an estimate of your
location. This is why "aGPS" (assisted GPS) services are so popular:
by starting with a rough city-level coordinate fix through something
like cell-tower network location, you can reduce the amount of math
you have to do to lock on. This is where the barometer comes in.
The 3 dimensions in space are latitude, longitude... and altitude. The
barometer gives you a reasonable first-cut estimate for altitude. This
gives you a bit of a leg up on one of the dimensions -- especially
combined with "2D" aGPS -- which can help speed up lock-on in general.
Now of course, the barometer can also be used for things like, well,
determining atmospheric pressure (although I'm not sure it's really
weather grade.) But the main reason it's in your phone is to help with