I am having some trouble in reading and understanding the relationship between notes.

My teacher gave me a piece and although I was note perfect I had almost written a new song and the rhythm wasn't just quite correct.

If I have not yet grasped the basics of rhythm could this account for my difficulty in getting some pices together?

I would say so--how can the piece be together if you are playing in a random rhythm?

I speak as a percussionist--GET A METRONOME!! They are invaluable tools. Of course, should not be able to afford one, use an online one.

A technique my brother's teacher taught him was to stomp his feet slowly saying "QUAR-TER" and then quickly "EIGHTH-EIGHTH" to get the relationship that two eighths should fill up the time of a single quarter.

I think listening to a metronome will help with this (she personally doesn't want him using one, but I'm not a pianist so I can't help suggesting it)--use one that has a visual cue as well, a blinking light would be best. Start very slow and then gradually increase the speed. This will help you adjust to any speed while still getting the rhythm right. While gradually increasing the speed, gradually decrease the volume too. Eventually, go just along with the visual cue, and then turn that off as well.

There are also a lot of rhythm books out there, but I have no recommendations on those (others will chime in with those).

Another suggestion I've seen in books, on the forums, and from teachers is to CLAP/TAP the rhythm before hand, first with just the right hand, then the left, then together, then with fingers.

RTC,
This is what I am working on now. I understand in theory how the rhythm works but cant play it.
I decided to put aside all the songs I was working on until I get this down.

I think the music note relationship chart is helpful. http://images.google.com/imgres?imgurl=h...l%3Den%26sa%3DN

Get a book or samples of lines of rhtyhm (it does not have to be on a staff-just a straight line of different notes seperated by barlines. and like II said use the metronome and say TA or DA to the rhythm. Your teacher can let you know if you are counting right.

I think its very important to do this and work on it now. Too many of my teachers in the past overlooked my rhythm problems and it is just as important to get the rhythm right as it is to play the right notes.

I don't actually have this book but I looked through it and I looks like it explains things very clearly (maybe I will order it myself...)
http://www.amazon.com/gp/reader/0793573793/ref=sib_dp_pt#reader-link

Good luck and keep us updated.