They made extensive use of snitches (Inoffizielle Mitarbeiter - IM), which were either intimidated or rewarded with easier access to universities or luxury goods normal "socialist brothers and sisters" had to wait for a long time.

In general they used a lot of manpower for their activities, so they had lots of photographing and sound recording technology, but mostly what you would expect: bugs, wiretaps, small and silent cameras, cameras to make photos around a corner. Telephones were rare and telegrams were used mostly for communication, so they were wired though the MfS headquarters so they could just print out everything going through.

The Stasi opened a lot of letters, not only from/to west germany but also in the DDR itself. They even had a "Handschriftenspeicher", a database with lots of writing probes, so if a antisocialist letter was found they could compare the handwriting with that of thousands of citizens and find out who wrote it.

They even collected smell sampled in jars, so a trained dog could be used to check who had been in a room, although it's doubted this method was reliable or efficient.

the german wikipedia article has some photos and desciptions of the used equipment (http://de.wikipedia.org/wiki/Ministerium_f%C3%BCr_Staatssicherheit#Ausr.C3.BCstung)