Basically CLR is Microsoft implementation of the the CLI standard in a nutshell. All you need to know is that if you are using Windows and programming with C# or VB.NET,etc you are using it and the services it provides for your programs:
The virtual machine aspect of the CLR allows programmers to ignore many details of the specific CPU that will execute the program. The CLR also provides other important services, including the following:

* Memory management
* Thread management
* Exception handling
* Garbage collection
* Security

CLR = Common Language Runtime; as daviangel explained, this is Microsoft's implementation of the CLI
CLI = Common Language Infrastructure; this is the ECMA standard that describes the .NET implementation (the virtual machine and runtime environment etc)
CIL = Common Intermediate Language; the bytecode to which .NET languages are compiled and which is JIT (Just in Time) compiled when the .NET code is executed
CLS = Common Language Specification; part of the CLI specification that specifies interoperation details of compliant languages.

Google will give you the gory details (the ECMA standards are available at no cost online) if you're interested.