When applying for a PhD, in the UK, it's almost certain you will need to write a research proposal. This proposal should be something you discuss with a potential PhD supervisor and should be something clearly aligned with their existing interests.
The proposal should:
- Situate itself within the current literature, citing several papers (at least 5) that are most relevant, as well as other papers in the general field. This should ideally include at least one paper from your proposed supervisor as it will then show why working with that supervisor makes sense for you and them;
- Demonstrate that you have read and understood (some of) these papers you are citing, and discuss their implications for your proposed work;
- Propose a number of research questions and potential directions for tackling them in your PhD. The more specific and concrete the better, but sometimes at this stage it might only be possible to point to a general direction of inquiry. Some high-level grand goals can be fine as long as there is a recognition that the PhD won't necessarily reach that goal, but will be able to contribute in that direction by making some key steps;
- Outline possible approaches for conducting the research: methods and techniques that could be used to attack the problems. This might include a sketch of some potential solutions, or at least an outline of a plan of what you could attempt first;
- A discussion of how you will evaluate your work. What are the kind of probes/tests that could be applied? These might be quantitative and comparative in nature (e.g., benchmarking, performance analysis, experimental), or more theoretical (e.g., proofs of key properties).
- Consider whether there are any ethical implications of your research or aspects that need thinking of in terms of responsible innovation, e.g., how will users of this work understand and build from it safely and appropriately.
- An optional extra could be a timeline of the work, though I usually find that it's so hard to accuately estimate at this stage that just having a split of work into 3-5 "packages" is more helpful.
It is quite common for PhD students to end up going in a different direction with their actual PhD research, rather than sticking to their proposal. The objective with the proposal is not to map out exactly what you will do, but to demonstrate that you have some sense of where you might go and how you might get there. It is to demonstrate some early skills in research (e.g., reading papers, synthesising questions and ideas) and your enthusiasm and interest in the subject. Another key function of the proposal is to make sure there is good alignment between the potential supervisor and student so that you will be able to benefit from relevant expertise.
You should send your proposal to prospective supervisors, suitably tailored, and ask for comments. Hopefully they will then provide high-level guidance and an assessment of whether they think your work would be a good fit for them. Alternatively, they may then suggest someone else who could supervise the work better.
It is usually best to prepare the proposal using a technical writing style similar to publications in the relevant field, e.g., using appropriate terminology, prose style, and citation format. For those writing proposals in mathematics, engineering, and sciences, using LaTeX to prepare the proposal is likely going to be a positive, demonstrating a useful skill that will be needed in the PhD, and producing a professional looking proposal.