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1. Name & address of company approached.
2. Contact details: Name of manager, email and telephone number.
3. Date I applied for job.
4. Other information: which allows me to records if I get a reply email or letter etc.
You could do that with MySQL, but it would be better suited to a spreadsheet because there is only one of everything for each row.
Now, if you were submitting several job applications to the one company, or to several people within that company, or to an agency which was recruiting for several companies, then you would need a Relational Database (MySQL) for that.
Imagine an Address/Phone book.
You could have several people at the one address (a family). In that family, each might have a mobile number, and a home phone number, and an email address (or two!) or some other contact info.
So you would need a people list, an address list (referenced by people list), a phone list, and people to phone brige table for many-to-many relation (many people share home phone, but have individual mobile), and so on.
Where everything is one-to-one, like your task, use a spreadsheet.
Where there are one-to-many and / or many-to-many use a database.
Good luck,
Barry.