UPDATE: This article describes the situation as it was in May 2015 before the transfer of territories took place.
First have a look at this news report stating that India’s parliament has finally passed legislation regarding the ratification of the land boundary with Bangladesh: http://www.thehindu.com/news/national/rajya-sabha-passes-bill-to-ratify-lba-with-bangladesh/article7177102.ece
So you are wondering what the fuss is about, and why it could not be resolved since 1947. First let us revise some basic geography. As usual, Wikipedia is a good starting point: http://en.wikipedia.org/wiki/Enclave_and_exclave
So you now know the difference between an enclave and an exclave, and that there are Indian exclaves in Bangladesh and Bangladeshi exclaves in India. The Bangladeshis would refer to the first type as enclaves in their country while the Indians would refer to the second type as enclaves in their country. These are also called first-order enclaves/exclaves.
So far so good. Then there is the counter-enclave or enclave within an enclave (or exclave within an exclave if you prefer). There are quite a few of these, namely an Indian exclave in Bangladesh which includes a Bangladeshi exclave totally surrounded by the aforementioned Indian exclave. Similarly you would have a Bangladeshi exclave in India which includes an Indian exclave fully surrounded by the aforementioned Bangladeshi exclave. These are known as second-order enclaves/exclaves.
Is your head starting to spin? Finally, we end with the counter-counter enclave or third-order enclave which does have one example. You might as well look at this on the map: http://en.wikipedia.org/wiki/Indo-Bangladesh_enclaves#/media/File:Dahala_Khagrabari.png
Cooch Behar has sometimes been the butt of jokes, with Rushdie’s “Midnight’s Children” having a minor character named the Maharani of Cooch Naheen (i.e. “The Queen of Nothing”). An old map showing the messy border of Cooch Behar can be seen here.
It can be enlarged: http://en.wikipedia.org/wiki/Indo-Bangladesh_enclaves#/media/File:Coochbehar.jpg
For an overall view, you could look at this segment from Google maps. Start from the quaintly named village of Phansidewa (“gallows”) and try tracing the border eastward through Jalpaiguri, Cooch Behar and finally Dhubri district: https://firstname.lastname@example.org,88.6693472,11z
A good overall summary is here http://en.wikipedia.org/wiki/Indo-Bangladesh_enclaves Note the apparently true story about the rulers of Cooch Behar and Rangpur exchanging villages as gambling stakes (though they really should have used something like poker chips). The article gives a full list of enclaves (though the actual documents run into hundreds of pages).
Life in these exclaves can be harsh. You would generally live in close proximity to trigger-happy border guards and barbed wire, while your “parent” country would not be able to ensure any services to your territory as you can see here: http://archive.thedailystar.net/2004/06/17/d40617070171.htm
This weird border has attracted the attention of trivia-hunters and serious scholars from around the world. Apart from various magazine articles referenced in the Wikipedia article, you can even download a doctoral thesis (over 500 pages long) with the quaint title of “Waiting for the esquimo”: “Waiting for the esquimo: An historical and documentary study of the Cooch Behar enclaves of India and Bangladesh”
Tail piece: My old friend Milan Chatterjee wondered if any railway line on the Indian side of the border crossed any Bangladeshi exclave. A closer look at the route from Cooch Behar to Gitaldaha and Bamanhat revealed that this was not the case, but that the abandoned line from Bamanhat to the border did momentarily cross three of these enclaves: https://www.google.co.in/maps/place/Cooch+Behar,+West+Bengalemail@example.com,89.5882181,14z/data=!4m2!3m1!1s0x39e2fc1ce181e4bd:0xe7787e092217f629 These stretches would total about 700 M (estimated by eye).
It is also understood that the old Gitaldaha Junction was relocated to a new location further from the border, and this was naturally known as New Gitaldaha Junction. It is still possible to visit the “old” Gitaldaha junction where a now-defunct bridge crossed the river into East Pakistan/Bangladesh.
One structure can be seen here. It appears to be right next to the border fence. http://www.distancebetweencities.co.in/kolkata_west-bengal_and_gitaldaha-ataiabari_west-bengal/gallery However, on the Bangladeshi side of the border the line from Burimari (on the border) to Lalmonirhat does pass through a few hundred metres of two Indian exclaves. It is not known if there is any particular problem with train operations here: https://firstname.lastname@example.org,89.0624778,14z
Finally, here is a 28-minute video by an Indian TV channel showing various aspects of life in the enclaves (Mainly English, some Hindi and Bengali): https://www.youtube.com/watch?v=e7hijHa0DYQ