Let me start with the obvious: the beaches.
Rio has the most beautiful beaches i've ever seen and they are of the main touristic point of the city - yes, we have Sugar Loaf an Corcovado too but i'll talk about them later.
They are not just beautiful, they are different. And the reason is the people. People from Rio (the cariocas) make the beaches a special place. It's a place to meet people, where you have special friends (the beach dudes) and where everybody goes every sunny weekend. Every summer weekend nobody does anything before going to the beach and Monday is the day to show the tan around.
The most famous beach is Ipanema - the one of the picture. It's my favorite neighbourhood and a wonderful place to live. It's just not very cheap but the place everybody should stay when visiting Rio.