Looking for translations? Try our unofficial guide to decoding British degrees.
A DPhil is what Oxford and some other British universities call a PhD. PhDs in Britain are different from those in the US: they are usually just the doctoral dissertation, without coursework, comprehensive examinations, opportunities to TA, etc. You must (almost always) apply with a clearly-defined research proposal and a prospective supervisor for your research. If you're offered a place, you conduct research guided by your doctoral supervisor, write it up, and usually do an oral defense. It tends to take three or four years to earn a doctorate.
Note: If you're hoping to go on to an academic career you should talk to a professor in the relevant field at home: since doctoral programs in different countries can be very different indeed it can be difficult to get a teaching job in some fields with a degree from elsewhere. So be sure to consult experts in your field.
Any degree beginning with "M" is a master's degree: MPhil, MSc, MA, MSt, etc. Most master's degree courses in Britain and Ireland take one to two years.
The title of a master's program alone may not help you determine whether or not it is a research or a taught course: always check the fine print...
Start by finding the universities and departments which seem to be best for what you want to study (see "Ranking universities," here). Check the admissions website/graduate prospectus of each relevant university to see what degree courses are offered there. Look at the syllabi, the reading lists, the lists of scholars working in that department and what they're working on, the research centers for the study of various topics, and any other resources there which might help to make a particular place the right one to pursue your specific area of interest. Beyond that, consider the following:
Talk with your professors and other mentors in the relevant field. They are the best people with whom to talk about graduate study. They are experts in their fields, they know the shape of current research, who's doing the most interesting work, and where the strong degree programs are. As your teachers and mentors, they also know something about your particular interests and strengths and what might be the best next step in your education.
Faculty in the relevant academic discipline are, moreover, the best people to talk to about finding a research supervisor and putting together a strong, interesting proposal—both of which are essential for application for a research degree.
Tip: Do some homework first, and your conversations with faculty and other mentors will be more fruitful. You'll not only get invaluable advice, but these conversations will also help your mentors to write stronger, more specific letters of recommendation. (Read how best to ask for letters and to prepare for an informational interview with a mentor.)