A library is a collection of precompiled routines that a program can use. Libraries are pretty handy when these routines are frequently used, since you do not need to link these routines to every program using them.
BigML bindings allow you to interact with BigML.io, the API for BigML. You can use them to easily create, retrieve, list, update, and delete BigML resources (i.e., sources, datasets, models and, predictions). The Binding term generally refers to a mapping of one thing to another and provide you with the API glue code to use BigML service in a particular programming language.
A Software Development Kit (SDK or "devkit") is a set of software development tools that allows the creation of applications for a certain development platform such as a software package, software framework, hardware platform, computer system, operating system, etc.
As you can see, all these three terms are pretty similar and may relate to the very same thing. In the BigML case, you can think of our API bindings as libraries that constitute a Software Development Kit for Machine Learning applications that use our cloud platform.