I want to know if it’s possible in the womb for a chromosome to break off and attach itself to another chromosome? If so, what would be the effects, and why would it happen?
Such an event is called a chromosomal translocation. I assume it happens because of naturally occurring DNA damage that sometimes isn’t repaired properly. The effects can vary. Some people who have translocations in all of their cells are fine, but have children with severe birth defects because of the way the altered chromosomes are inherited. It’s kind of hard (at least for me) to explain without diagrams, but if you do a search for chromosomal translocation on Google you might be able to come up with a good page on the subject.
By “in the womb”. I suspect you mean to a developing foetus? This type of thing happens rarely, and happens in single cells at a time, which then proceed to divide, if they are viable. If the foetus is already developing, then it would only affect part of the child’s body. I would think, though, that this type of event would more likely result in death of the particular cell, or in a tumour, which might lead to a miscarriage or stillbirth.
If this happened before conception, i.e. in the sperm or egg, then there are a variety of possible diseases, Down’s Syndrome being the most well-known. IIRC it’s a translocation of the 21st chromosome onto the 16th, but don’t quote me on that 
More than you want to know coming up:
Indeed it is a chromosomal translocation. Specified by t(x;y) as in t(16;22) where x and y are the two translocated chromosomes.
Down Syndrome can result from a translocation, but it is a special case of Down Syndrome which can be heriditary (it depends on a Robertsonian translocation). Usually, Down Syndrome is caused by a defect in the mitotic spindle of the developing egg, leading to a phenomenon of three independent chromosome 21s. This is called nondisjunction.
Most translocations that you would see as congenital abnormalities (i.e. in the womb) actually occur in the egg or the sperm. This would lead to every cell of the developing fetus carrying the translocation. There are hundreds of translocation syndromes described (a search for “chromosomal translocation” in OMIM brings back 447 hits). They usually involve some kind of dysmorphism, often times mental retardation. Interestingly many translocations have no phenotype, and can be a cause of infertility later in life (as the chromosome pairs don’t match up right upon fertilization, with parts of chromosomes only being represented once and other parts being represented three times). Again, hard to describe without figures.
Lastly, chromosomal translocations can occur at any point during life. Several very well described translocations are correlated with cancer, for instance t(9;22) is called the Philadelphia chromosome and is correlated with chronic myelogenous leukemia.