Moncloa is a nice area for students as is nearby Plaza de España. La Latina and Lavapiez are areas or "barrios" full of a mix of people and cultures, and can be a bit new for some people. Malasaña is a brilliant student/artistic area full of young people. Areas like Salamanca and Chamartin are a bit more high end, Salamanca has the famous shops like Jimmy Choo and Armani on the same roads and Chamartin is almost the financial businessy area, it has some nice places to stay and eat but it's more somewhere to go if you're on a work trip.
I spent a year living here as part of my course at uni and I absolutely loved it. Best time in my opinion to go is between March and June. A lot of the city starts to undego building works over the Summer when everyone is off on holiday/ not working from the heat. I know for a fact my lab closes for the whole of August from the heat. It's a very laid back city, very relaxed, full of bars and lovely people. Personally I struggled finding a lot of touristy things for friends and family visiting to do here but it's a great place for students and art lovers. It truly is a beautiful city. Though English travellers beware, they are very laid back about eveyrthing, especially timings!
Very pleasant city in the south of Spain. There is a city beach, many touristic sites, good and affordable restaurants with delicious food. Many places are free to visit at Sundays.