1. To build or to create something again which has been damaged or destroyed so it will work again.
2. To change a system or an organization completely, so it works more effectively or in a different way: "The committee was given the responsibility of reconstructing the city's public transport system."

If anyone reconstructs anything which has happened in the past, he or she will have to combine a lot of small pieces of information to get a complete concept of what happened: "The police have been trying to reconstruct the crime by using the testimony of many witnesses."