Hi! Could someone explain me what the difference between 入る「はいる」・入る「いる」and　入れる is?

It's been a while and my Japanese is a bit rusty, but here goes:
いる gets a lot fewer hits in my dictionary than はいる and they have a very similar meaning of 'entering' or 'going in'. However, はいる carries a whole series of further meanings e.g. 'to contain' , 'be included', 'be installed' and therefore you're much likelier to encounter it.
入る and いれる are also a 他動詞/自動詞 (transitive/intransitive verb) pair. There are vast vast numbers of these - the upside being that you can often learn verbs in a '2-for-1' fashion, the downside being that you have to remember which is which. There are rules about which particles you use for which as well (in the first instance a を/が although there are unfortunately more complex situations than that) - are you using a particular textbook? It should have an explanation of sorts - if you're just trying to learn from native material, I would definitely suggest that you look up the grammar patterns for transitive/intransitive verbs so that you understand what you're looking at, and then use native material to get a feel for which you should use when.

Anyway, back to the meanings:
私は本を箱に入れた。 I put a book in the box.　Transitive - the book has had an action done to it, takes the particle を and いれる is your verb of choice.

箱に本が入っている。 A book is (in the state of being) in the box. Intransitive - there's no object required, the book is just 'in' the box, takes が and はいるis the verb to use.

Does that help at all? It's been a few years since I studied this kind of stuff and after a while you sort of 'know' which one to use when just through exposure. Doesn't help in the beginning when you've got a teacher standing at the front of the class shouting '自動詞ですか？　他動詞ですか？’ and you're busy frantically particle-choosing to try and decide...!

There are many shades of difference, but はいる　is more like "enter" as in to enter a room. いる　is more like entering a tub, or to have in hand, or for a river to flow into a sea. いれる　is to insert something, such as a coin into a machine. As the example from clairehansell86 shows, if the book is in the box, that is いる, but to put it there is いれる. All use the same kanji 入.

Thank you both! Now I have all clear!

はいる is the modern form of いる which is seldom used as a standalone verb nowadays. The stem of いる is too short and can easily confuse with 居る, 要る, 煎る, 鋳る, 射る, etc. when spoken, so the stem is elongated with は and resulted in はいる. But, we can still find いる used in a lot of idiomatic expressions like
「気にいる」,「堂にいる」,「有卦(うけ)にいる」, etc.