PyZMQ uses different licenses for different parts of the code.
The 'core' of PyZMQ (located in zmq/core) is licensed under LGPLv3. This just means that if you make any changes to how that code works, you must release those changes under the LGPL. If you just use pyzmq, then you can use any license you want for your own code.
We don't feel that the restrictions imposed by the LGPL make sense for the 'non-core' functionality in pyzmq (derivative code must also be LGPL or GPL), especially for examples and utility code, so we have relicensed all 'non-core' code under the more permissive BSD (specifically Modified BSD aka New BSD aka 3-clause BSD), where possible. This means that you can copy this code and build your own apps without needing to license your own code with the LGPL or GPL.
When you contribute to PyZMQ, your contributions are made under the same license as the file you are working on. Any new original code should be BSD licensed.
Examples are copyright their respective authors, and BSD unless otherwise specified by the author. You can LGPL (or GPL or MIT or Apache, etc.) your own new examples if you like, but we strongly encourage using the default BSD license.
Some code outside the core is taken from other open-source projects, and inherits that project's license.