A dynamic library is usually identified by a .so (for "shared object") suffix (e.g. libc.so). Like a static library, this kind of library also contains the modules that you want to include in your program, but these modules are not bound to your program at link time. Instead, your program is linked in such a way that the Process Manager causes your program to be bound to the shared objects at load time.
The Process Manager performs this binding by looking at the program to see if it references any shared objects (.so files). If it does, then the Process Manager looks to see if those particular shared objects are already present in memory. If they're not, it loads them into memory. Then the Process Manager patches your program to be able to use the shared objects. Finally, the Process Manager starts your program.
Note that from your program's perspective, it isn't even aware that it's running with a shared object versus being statically linked — that happened before the first line of your program ran!
The main advantage of dynamic linking is that the programs in the system will reference only a particular set of objects — they don't contain them. As a result, programs are smaller. This also means that you can upgrade the shared objects without relinking the programs. This is especially handy when you don't have access to the source code for some of the programs.