Where legalities are concerned, you really should be asking a solicitor. That said, however, our understanding is that you basically own half the thickness of any party wall - so as far as minor, everyday things are concerned,
like drilling into it to install fixings for kitchen cabinets or shelves, or even cutting into it to install electrical wiring and sockets, you are basically free to do what you like. That said, it is no more than common courtesy to let your neighbours know before you start.
Major structural work, on the other hand, is another matter, and is governed by the Party Wall Act, 1996, most recently updated in January this year. Basically, this enshrines your right to do things like cutting into the wall to take one end of an RSJ, or slicing all the way through it in order to install a new damp course. If necessary, it even permits you to demolish the whole thing and completely rebuild it. You must however give the owners of the adjoining property written notice, not less than two months ahead of the planned start date. This notice should be dated, and it is advisable to include a clear statement that it is a notice under the provisions of the Act. Your neighbours should in turn give their consent - again, in writing - within 14 days.
When it comes to the work itself, the Act stipulates that it should not cause undue inconvenience to your neighbours. It also makes you legally responsible for providing suitable protection for buildings and property during the course of the work, and for compensating your neighbours in the event of any incidental damage. In return, your neighbours are legally bound to allow free access to their property as necessary for the proper completion of the work.
Finally, whilst our understanding is that a neighbour can't actually stop you from exercising your rights under the Act, failure to do things by the book could trigger a lengthy and potentially costly dispute. And disputes can happen all too easily, since all the neighbour needs to do is to trigger one is to withhold his or her written consent. That's why I would strongly urge you to discuss your plans with your neighbours well in advance - and to seek proper legal advice!