Legal writing is all of the writing that lawyers must do throughout the course of their job. This can involve writing wills, contracts, briefs, memos, etc.
Legal writing often involves the use of many legal terms, so it requires a lot of legal skill and legal education. For this reason, people without legal training usually cannot do legal writing.
A lawyer can have to do an extensive amount of legal writing during the normal course of their work. For this reason, many lawyers hire paralegals to help them. Paralegals can do a lot of the legal writing, but lawyers will usually do the most important parts themselves. For example, a lawyer may allow a paralegal to do a case briefing, but not set up an official contract between two parties, which will be legally binding.