In a nutshell
Binary addition adds two numbers column by column, just like decimal addition, except each column holds only 0 or 1 and may pass a carry to the next. A ripple-carry adder builds this in hardware by chaining one 'full adder' per bit, starting from the rightmost column. It's simple but slow, because each column has to wait for the carry from the column before it to ripple all the way across.