So the basic purpose of a library is to allow code that does some useful thing to be easily used in multiple programs. Like say math functions beyond what is in the language it self or creating network connections.

When you build a program with multiple source files there are many steps. First each file compiled into an object file. This is machine code but wherever you have calls into other files it just inserted a note that basicly says connect this call to this part of another file. So for example connect this call to SquareRoot function in Math library.

After that has been done to every file needed then the linker steps in. It grabs all the object files combines them into one big file and then looks for all the notes that say connect this call to that function and replaces them with actual calls to the address where it put that function.

That is static linking. All the code ends up in a big executable. Simple but it has two big problems. The first is size. Doing it this way means every program that takes the squareroot of something has a copy of the entire math library. This adds up. Second is if there is an error in the math library every program needs to be rebuilt for the fix to apply.

Enter dynamic linking. With that the linker replaces the note to connect to the SquareRoot function in math library with code that requests the connection be made by the operating system.

Then when the program is run the OS gets a list of the libraries needed by the program, finds them, copies them into the memory reserved for that program, and connects them. These are .so files on Linux and .dll on Windows.

Now the os only needs one copy of math.so and if there is a error in the library a update of math.so can fix all the programs that use it.

For GPL vs LGPL this is an important distinction. The main difference between them is how they treat libraries. (There are other differences and this is not legal advice)

So if math.so is GPL and your code uses it as a static link or a dynamic link you have to providd a copy of the source code for your entire program with any executable and licence it to them under the GPL.

With LGPL it's different. If math.so is staticly linked it acts similar to the GPL. If it's dynamicly linked you only have to provide the source to build math.so and licences it under LGPL. So you don't have to give away all your source code but you do have to provide any changes to the math library you made. So if you added a cubeRoot function to the math library you would need to provide that.