The phrasal verb 'take on' has a few different uses depending on the context.
If you 'take on' something it can mean to accept a particular task, to agree to do an amount of work, or to be responsible for something. So you can 'take on a new client' or 'take on additional responsibilities', for example. You can also 'take on staff' meaning to hire them. In this way 'to take on' is similar to the verb 'acquire' or 'to accept' something or someone.
To 'take on' something or someone can mean to compete against an opponent or to put yourself against the competition. So you can 'take on a rival' or 'take on an opponent'.
You can also 'take on' on certain characteristics or behaviour when you demonstrate them to others. You might 'take on' the voice of your mother if you were impersonating her, or 'take on' an authoritative tone if you need to appear as an expert.
Additional examples with 'take on' are:
When you take on a client you begin to work for them.
Are you in a position to take on more work at the moment?
I have heard that the Hotel in town is taking on staff, you should apply for a job.
I think it's time we took on another employee to help us with the workload.