Educational and enjoyable - a great combination
(Recommended)
Pros
Free to visit, although it is nice to leave a small donation if you really enjoyed your experience. The museum is absolutely stuffed full of everything you could possibly want to know about the natural world and is a fantastic visit for children and adults alike.
The building itself is an amazing work of architecture and the gardens are lovely to sit in and relax during the summer. If you live in London and are looking for a cheap day out why not take some paper and a pencil and go and draw some of the exhibits (or the building itself?).
Excellent collection of dinosaurs and fossils and the blue whale has to be seen to be believed (even then it's quite hard to believe the scale of the thing). Children will also enjoy the earthquake exhibit in which you can experience what an earthquake might feel like. Temporary exhibitions are normally excellent. Allow a full day for visiting this museum; even then you may struggle to get round it all.
Cons
Can be very busy at weekends, especially the more popular exhibits like the dinosaurs. Sometimes full of school trips. Avoid this by trying to keep visits to weekdays and hoping there's not loads of school trips there that day!
Cafes and restaurants are quite expensive, although this is to be expected given the location and the nature of the business. Coffee and a muffin isn't too expensive but the sandwiches are! You can take packed lunches to eat in the picnic areas or in the gardens.
Occasionally the gardens are closed for events like London Fashion Week.