If you bring a legal case, you will be referred to by a specific title that depends entirely on which procedure you are using. Getting the terminology wrong in a document is not usually fatal, but it signals that you are not familiar with the process.
Claimant
Used in civil proceedings under the Civil Procedure Rules (England and Wales, from 1999). You are the claimant; the person you are suing is the defendant. This replaced "plaintiff" under the Woolf reforms.
Plaintiff
The old term, still used in some older cases and in other jurisdictions (including the United States and older Commonwealth countries). In England and Wales, if you see "plaintiff" in a case decided before 1999, it means what "claimant" means today.
Applicant
Used in applications to a court or tribunal where you are seeking an order rather than a money judgment — for example, in family proceedings, immigration tribunals, or judicial review. The other party is usually the "respondent".
Petitioner
Used in specific proceedings such as divorce (historically), winding-up of companies, and some insolvency matters. Again the other party is the respondent.