You need to learn about phrasal verbs:
http://grammar.ccc.commnet.edu/grammar/verbs.htm#phrasal (Be sure to click on the word HERE to see very useful lists of these verbs.)
and
http://www.eslcafe.com/pv/
And here are several dictionary definitions of this particular phrasal verb, "break up":
http://www.google.com/search?hl=en&rlz=1C1GGGE_enUS379US379&q=define+break+up&oq=define%3A+break+up&aq=0&aqi=g1g-v2&aql=1&gs_sm=c&gs_upl=9087l9762l0l12420l3l3l0l0l0l0l281l753l0.1.2l3l0
Let me know if you have more questions about phrasal verbs or this phrasal verb in particular.