You will want a laptop with at least 4GB of RAM (this will be enough for most tasks) and any processor from Intel or AMD (apart from the Intel Celeron range, stick to the i3/i5/i7s).
With student laptops the limiting factor on speed is going to be the hard drive. We recommend that you look specifically for a hard drive that runs at 7200 rpm (if the drive does not state this explicitly, it is probably running at 4800 rpm, and will slow down within 1 year). We recommend avoiding drives larger than 512GB, as they tend to be slower.
If you have a slightly larger budget, a much better option than a mechanical hard drive is a solid state drive, which has no moving parts, and thus last much longer, and is much much faster.
Laptops with these specifications and a mechanical drive start at around £350, while the solid state options start at £450.
If you want a mac instead of a PC, this will cost a lot more, starting at around £900. If you choose to take this route, ordering the machine from inside the university network will allow you to have a 10%-15% student discount. The IT office is happy to help you with this.