The phrasal verb ‘break up' has a number of different meanings in both professional and social contexts. To ‘break something up' can mean to separate or divide an object into smaller parts. If you ‘break up’ an object, you disassemble it. You can also ‘break up' a group of people, meaning to divide them into smaller groups. If you 'break up with someone', you end a romantic relationship. So, you could 'break up with’ your boyfriend or girlfriend, or 'break up with’ your partner. This leads to the noun 'a break-up’.
If a party or an event 'breaks up', it comes to an end, and the people in attendance begin to leave. You can ask someone, 'what time did the meeting break up at?', meaning 'what time did the meeting end?'
When a school, university, or business formally stops work for a time, you can say they 'broke up' for the holidays or 'broke up' for the midterm. When you are on the phone with somebody and have a bad connection, you can say they are 'breaking up', meaning it is difficult to hear them or to make out what they are saying. If a person 'breaks up', it refers to a moment when they use control of their emotions, usually due to being extremely upset or laughing uncontrollably.
Additional examples with 'break up' are:
A radio DJ is telling the listeners about the history of a band:
"The singer's alcohol problem lead to the band breaking up in 1998."
The news is reporting on a protest happening in the city:
"The police arrived and attempted to break up the protest after violence broke out."
A group is faced with a difficult problem. Someone suggests dividing into smaller groups to search for a solution:
"We should break up into different working groups to consider the problem and get back together at the end of the day."