What's the purpose of the LEA instruction?

As others have pointed out, LEA (load effective address) is often used as a "trick" to do certain computations, but that's not its primary purpose. The x86 instruction set was designed to support high-level languages like Pascal and C, where arrays—especially arrays of ints or small structs—are common. Consider, for example, a struct representing (x, y) coordinates:
struct Point
{
     int xcoord;
     int ycoord;
};
Now imagine a statement like:
int y = points[i].ycoord;
where points[] is an array of Point. Assuming the base of the array is already in EBX, and variable i is in EAX, and xcoord and ycoord are each 32 bits (so ycoord is at offset 4 bytes in the struct), this statement can be compiled to:
MOV EDX, [EBX + 8*EAX + 4]    ; right side is "effective address"
which will land y in EDX. The scale factor of 8 is because each Point is 8 bytes in size. Now consider the same expression used with the "address of" operator &:
int *p = &points[i].ycoord;
In this case, you don't want the value of ycoord, but its address. That's where LEA (load effective address) comes in. Instead of a MOV, the compiler can generate
LEA ESI, [EBX + 8*EAX + 4]
which will load the address in ESI.
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