1.  How many movements are there within Judaism?

Generally speaking there are 4  predominate "movements" within Judaism.
Orthodox, Conservative, Reform and Reconstructionist.  The differences are identified more by their interpretation and observance of the Mitzvot.  Each movement issues "possition statements" on certain topics.  These are not binding upon anyone but are considered to be the continuation of our on-going interpretation and application of Torah.  While the Reform and Conservative have a central headquarters which issues "Responsa" to issues, none of these are binding upon the members. Each community is free to participate as they see fit. For example, if a Reform Rabbi does not wish to perform an Inter-faith marriage, he/she is not obliged to do so - even those the Reform Movement allows for it.  Same with Conservative and Reconstructionist.  

2.  What are the most recognized/predominant ones?

Physically recognizable of course would be the Hassidim which fall under Orthodox - these are the 'men in black' - You might consider them the "pentecostals" of Judaism.  They are very strict, and they are very expressive in their service (not something one might think.)  There are roughly 15 million Jews world wide.  In the US, the top two movements by  membership are Reform closely followed by Conservative.  The smallest is the Reconstructionist.  There is another minor group called Karaites.  This group  observes only written Torah (e.g.  they eat meat and dairy together, they don't light candles for Shabbat etc...).  They may be smaller than the Reconstructionist.  I only know of one community in San Francisco.  Karaites are the continuation of the Saducees who rejected the notion of life after death, oral traditions (Talmud)  etc.

3.  How do they work together?

There is the old story of two Jews stranded on a desert island. They work very hard and very well with each other to build three synagogues.  One for each and one nither would ever set foot in.  

In reality, most cities that have a reasonable Jewish Community have a "Jewish Community Center" (or something like it).  This is the common ground where everyone comes together as Jews to support each other and engage in social action.  

4.  Are any movements rejected by the larger body?

The largest body is Reform - they respect the Orthodox and Conservative.  The Orthodox, have on occasion, been very negative towards the others.  But then every Jew knows that the Hassidim (the most observant) welcomes any jew of any - or no - affiliation to any of their services.  Their services are open to any Jew.

5.  What are the major/minor differences between the Orthodox, Reform and Conservative camps?

Briefly:  Up until 200 years ago, all  Jews were Orthodox (to some degree or another).  

Reform Judaism sprang up in Germany in the 1800's.  Its purpose was to change the Gentile attitude towards Jews.  Jews wanted to be considered Germans of the Jewish Faith (like Germans of the Lutheran, Catholic etc. faith).  They even attempted to hold services on Sunday rather than Saturday.They dropped Hebrew etc.  The goal was to stop oppression.  Well, that didn't work did it?   The Reform Movement says observance to the Mitzvot should be personal.  In other words, don't fret about observing them all today, pic one, make it permanent and work on adding in the rest.  Over the past 10 years, there has been a movement at the Reform Yeshiva (Rabbinical schools) to return to tradtion and observance of Mitzvot.

Conservative Movement arose in the US in rejection of the Reform Movement.  The Conservative Movement stresses observance, but also accept that it must be adjusted to meet the needs of the present time. For example, Orthodox would never drive on the Sabbath.  The Conservative says - our US culture oftentimes demans that we use a car on Sabbath.  So use the car to get to service - but don't stop by the Mall on the way home.

Reconstructionist sprang out of the Conservative Movement.  They are the most liberal - even to the point of taking no position on G-d being a 'person', and allowing for the possibility that the Torah was developed by man over time.  

***sometimes G-d loves variety and smiles***

I could not agree more.  Just look at creation!


Best to you and thank you for the questions.  
Shalom