Strictly speaking, this is really one for a solicitor. That said, my 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 in essence free to do what you like, as long as you don’t do it in the middle of the night! That said, it is of course no more than common courtesy to let your neighbours know before you start.
Major structural work, on the other hand, is a rather different kettle of fish. This is governed by the Party Wall Act, 1996, which 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, you can even demolish the whole thing and completely rebuild it! However…you must inform the owners of the adjoining property, in writing, not less than two months ahead of the planned start date, and they must give their written consent within 14 days of receipt.
The Act also stipulates that any work done should not cause undue inconvenience for your neighbours. In addition, it 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, while my 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 need do to trigger one is to withhold his or her written consent. That’s why I would strongly urge you to seek proper legal advice before doing anything too drastic.