Hadrian's general approach to the Imperium was that of consolidation. He opposed the continuation of Trajan's expansionism (possibly assassinating expansionist senators early in his reign) and sought to better prepare the army for defensive readiness. He was a military man, occasionally eating and sleeping with the troops. This put him at odds with the Senate and, alongside his general love of Hellenic culture, made the political climate stormy throughout his lifetime. Hadrian's legal reforms, including changing the status of the Italian provinces and the creation of the Perpetual Edict (effectively creating a bureaucracy), brought him further into conflict with the Senate.

Whilst you're trying to avoid military discussion, it might be of interest for other people to quickly race through a few of them:

1. Hadrian ordered few military operations. He was more interested in maintaining rather than expanding.

2. He surrendered lands won by Trajan in Mesopotamia.

3. He relaxed Roman control in Dacia and Parthia, allowing client kings to rule in his stead, maintaining peace through diplomacy.

4. He established Hadrian's Wall, built from stone on the northern frontier in Britain, and the wooden Danube fortifications, delimiting the frontiers of the empire.

5. He insisted on strict drill and training to keep the army at a high state of readiness despite operating in a largely passive role.

Outside of the military sphere, Hadrian was decidedly enamoured with culture, especially that of Greece. He rebuilt the Pantheon, constructed a vast and impressive villa, and had a deep personal interest in architecture - he was a determined and proactive builder, ordering the construction of libraries, bathhouses and theatres as he travelled. Even his facial hair, Roman men being mostly clean shaven, was a nod towards Greece and became quite fashionable, remaining so even after his death. Did I mention he wrote poetry in Latin and Greek? He was just that kind of guy.

Perhaps most interesting are Hadrian's attempts to foster unity throughout the empire. Ideologically, he wanted to ensure that the empire felt unified by Rome rather than oppressed under Rome. The "Panhellion", a proto-parliament, was an attempt to join together the various Greek city states - a noble idea but futile. His extensive travels to the corners of the empire combined with his policy of poised defence was a calculated move to reinforce the cultural underpinnings of a gigantic fractured state. Unfortunately, his grand gesture in the Levant - rebuilding Jerusalem under the name Aelia Capitolina - was met with stern resistance from the Jewish population. A rebellion broke out in 132 and ended in 135 after half a million Jews had died. Largely his rule was a peaceful one.

Unfortunately, the pressure building in the Senate would boil over as the issue of succession came to a head in the last two years of his reign. Dying in 138, after 21 years of rule, he was succeeded by Antoninus Pius who later had Hadrian deified. It had not been an easy succession with a candidate dying unexpectedly and at least one attempted coup being repressed. Hadrian managed to line up Marcus Aurelius as the heir to the empire after Antoninus. Marcus would be the last of the "Good Emperors".

There's a whole load more available on the internet, might I suggest here, here and, of course, Wikipedia. These sites go into more personal detail, including his lover and friend Antinous.

^Edit: ^SPAG

The biggest thing that I'm aware of is crushing the final Jewish revolt, (and renaming Judaea to Palestine) leading to the expulsion of the Jews from Judaea. He tried to get rid of all signs of the Jewish national identity from the area (even destroying the city of Jerusalem and building a new, Roman provincial capital on top). Essentially, he tried to totally disconnect the region from the Jewish culture. This is because the Jews had a strong national identity and had been giving Rome a lot of trouble, going back to the first Jewish Rebellion in the 60s AD (and inflicting the biggest Roman military defeat by a rebelling province, at Beth Horon). He felt by eliminating that cultural sense of "Jewishness" from the area he could stop any further revolts from the Jews.