What does it mean for an object to be picklable (or pickle-able)?

Things that are usually not pickable are, for example, sockets, file(handler)s, database connections, and so on. Everything that's build up (recursively) from basic python types (dicts, lists, primitives, objects, object references, even circular) can be pickled by default.

You can implement custom pickling code that will, for example, store the configuration of a database connection and restore it afterwards, but you will need special, custom logic for this.

All of this makes pickling a lot more powerful than xml, json and yaml (but definitely not as readable)

It simply means it can be serialized by the pickle module. For a basic explanation of this, see What can be pickled and unpickled?. Pickling Class Instances provides more details, and shows how classes can customize the process.

These are all great answers, but for anyone who's new to programming and still confused here's the simple answer:

Pickling is making it so you can store it long term and get it later without it going bad. A bit like Saving in a video game.

So anything that's actively changing (like a live connection to a database) can't be stored directly (though you could probably figure out a way to store the information needed to create a new connection, and that you could pickle)

Bonus definition: Serializing is packaging it in a form that can be handed off to another program. Unserializing it is unpacking something you got sent so that you can use it