Do you know the difference between random access and sequential access? Let’s ignore fragmentation for the sake of simplicity.
The former means accessing data which is scattered all over the hard drive; a typical example is what happens when you boot up your PC, or launch a particularly “heavy” piece of software: lots of different files need to be read, and they’re all over the place.
The latter means accessing bits of data which are one next to the other, e.g. copying a large file.
Think of a librarian who needs to get hold of certain books: random access is like getting one book from the kids’ section, one from the history section, one from fiction, etc. Sequential is like getting 3 books by the same author, all next to each other.
SSDs are faster in both tasks, but are way faster than ordinary drives when it comes to random access, which is why you’ll see the biggest speed improvement when booting up the PC or launching certain software like Photoshop. The difference in the bootup time for Windows 10 can easily be 15 seconds for SSD vs 45ish seconds for an ordinary drive. Some more info here: uk.pcmag.com/storage-devices-reviews/8061/feature/ssd-vs-hdd-whats-the-difference
To be clear, not every task requires reading data from a drive; e.g. calculations run entirely in memory are not affected in the slightest by the kind of drive you have.
A common piece of advice, which I agree with, is to get one SSD to install the operating system and the software you need, and another, cheaper and slower, hard drive for your data, music, videos, etc. Nowadays you can find a 128 GB SSD drive for ca. £45, and a £256GB for ca. £70-80. If you’re on a very tight budget you can of course save by avoiding an SSD drive, but in that case be sure not to overspend on other components, e.g. avoiding an SSD drive and then buying the latest, most powerful CPU would be nonsensical, unless you have very specific computing needs.
Online shops like PCSpecialist and CyberPowerPC can build PCs customising every single component to your choice. I have ordered from both in the past.