Equipment/Computer Budgets for CS PhD Students
This answer is based on my experience in France and Ireland. I would assume that it's similar at least in other European countries [please edit or comment if I'm wrong].

In computer science PhD students are usually provided with everything they need to do their work including (at least) a desktop computer, office space, and generally there is also a budget for traveling to conferences. Depending on the country they might have the status of staff member or not, but even if not the working conditions are very similar to those one would expect in a regular job (well, an academic job, don't expect too much!).

The budget related to equipment/travel is usually a part of the PhD funding which is given to the institution, as opposed to the main funding which is a grant or salary given to the PhD student. The budget for traveling/accommodation is usually the most limited part, it might even happen that there's no budget at all for this. It's a good idea to ask the supervisor about it at the beginning in order to submit papers to conferences in a sensible way: it's a shame to have a paper accepted but find oneself unable to attend the conference.

PS: in terms of computer equipment, in computer science it's also common to be provided with an access to some kind of high performance cluster.