If you want to build commercial software, use the Apache license version. It’s a more permissive license than LGPL, so you won’t run into any problems using Jackson under the Apache license in your proprietary work provided you follow every term of the Apache license (all are very reasonable). The main difference is that the LGPL might require you to make your application open-source under certain use cases and the Apache license will not.
Depending on your use case the LGPL version might also be usable but the Apache license is a safer bet.