A trading name is a name that a company or person trades with, rather than using their actual name. The acronym T/A is often used to mean ‘trading as’
For example, Bob Smith who is a self-employed carpenter might trade as Elite Carpentry; his full name in a business sense would therefore be Bob Smith T/A Elite Carpentry and any legal documents should encompass the full name to avoid confusion and add certainty. Similarly, a limited company could also trade under a different name; as an example there could be Carpets 4 Cheap Limited T/A Sofas 4 Cheap and again, it is important to provide the full name in any legal documents.
If you are entering an agreement with a business and you are unsure on who the actual contracting party is, then you should seek these details from the business.
Individuals and companies can have as many trading names as they want, however, they cannot use their trading name to try and pass-off that they are connected with an already established business with that name. If you have a trading name then you might want to consider registering it as a trademark so that other businesses cannot use it too – the Government has a handy guide for this process (https://www.gov.uk/how-to-register-a-trade-mark).