I'm new to open source, and I'm working on my first Java application, which will be released with the MIT license. I'd like to use Berkeley DB for object persistence. That product uses the GNU Affero General Public License. There seems to be a license incompatibility here, but as a non-lawyer newbie, I don't know how to proceed.
These two licenses are well-known to be compatible. What this means is that you can take codes under both licenses and combine them into a larger work. It does not mean that you are not limited in how you can license the larger work.
In particular, nothing prevents you from releasing your own code under the MIT license but the whole program (which combines the part that you wrote and the library under AGPL) will need to be released under AGPL only (since AGPL is strictly more restrictive than MIT).