I've heard about MongoDB, but I'm not sure I fully understand the concept.
If I have multiple JSON objects stored in MongoDB:
[{"id": "peter",
"age": "12",
"gender": "male"},
{"id": "gemma",
"age": "12",
"gender": "female"},
{"id": "paul",
"age": "13",
"gender": "male"}]
How would I be able to query all JSON objects with age >= 12?
First of all, understand that JSON is just a serialization technique. In and of itself, this serialization method probably should not determine your persistence medium. Looking at your question on the surface, it seems like what you are looking for is a typical relational storage database where you can use SQL to query against your data in a flexible manner.
Serializing/deserializing JSON data for storage into or for presentation after retrieval from such a relational database is trivial in pretty much any programming language.
Now if you truly need to store various snippets of JSON documents (or any other sort of document) that don't really have a fixed structure, that is really when you typically would start looking at a NoSQL type of solution such as MongoDB. One other possible such scenario for using the more popular NoSQL databases is when you are dealing with massive amounts of data and need to scale horizontally (i.e. the data is so large you need to scale the database across multiple servers). Many NoSQL systems make this much easier to do than traditional relational DB's. Of course in such a scenario, you would then need to evaluate those tools based on the functionality they provide in allowing you to read, write, and query data in the most useful manner for your use case(s).