There are no specific licences that you need to trade on eBay. In fact, eBay rules prohibit the sale of some of the items that would normally need a licence to sell in the UK, for example alcohol. You will, of course, need to be registered with eBay as a business seller before you can start trading.
Licensing of second-hand dealers
Local authorities in Scotland require second-hand dealers (including motor dealers) to obtain a licence or registration to operate. This applies unless dealing in second-hand goods is only incidental to the main business activity. Elsewhere in the UK, some - but not all - local authorities license or register businesses where second-hand dealing is the main or a significant part of the business and is not just incidental. However, certain specific exemptions generally apply - these typically include motor vehicle dealers and jewellers. If you are in any doubt as to whether second-hand dealer licensing may apply to your business, contact your local authority trading standards department for guidance.
The Money Laundering Regulations apply to High Value Dealers who are prepared to accept the equivalent of 15,000 euros in cash for any transaction, including where money is paid directly into your bank account. If you handle transactions of this size you must register with HM Revenue & Customs (HMRC) and put in place anti-money laundering systems.
Selling goods to overseas buyers
The Export of Goods (Control) Order requires an export licence to be obtained for antiques and art works over certain age and value limits if they are to be exported from the UK. Full details are available from the Art Council's Export Licensing Unit.
Export licences are also needed for certain other goods, although it's not particularly likely that you would be selling these on eBay.