“She got tied up at work.”

Your friend invited you and your wife out to dinner. Your wife couldn't come because she suddenly had something important that she had to do at work. You're explaining why your wife didn't come. You say:

She got tied up at work.

get tied up (with something)

To "get tied up with" something means to become really busy with something that you aren't able to stop doing. People usually "get tied up" at work. They get a lot of work that needs to be done quickly. People can also "get tied up" with their kids, doing home repairs, or other things:

Oh, is it eleven o'clock already? I got so tied up with sending out these invitations that I didn't even notice.

If you tell people that you're "tied up", they'll understand that you're too busy to do anything else. For example, if you call someone and they say:

Sorry, I'm kind of tied up at the moment. Can I call you back?

...you should understand that they're too busy to talk now.

at work

There are lots of different ways to talk about a person's job. Here's an explanation of when to use some of them:

• "at work" - use this when you're talking about something that's happening while someone is working:

  I got yelled at today at work.

  Mark's still at work.

• "in my job" - use this to talk about things that you have to do for work on a daily basis:

  I have a lot of problems that I have to deal with in my job.
  

• "at my company" - use this to talk about large-scale things that are happening to the organization of the company you work for.

  The management at my company seems to change every year.