The USAJOBS interface has done themselves no favors by implementing the drop-down menu shown in the way that they have. It is confusing, in that "Student (High School)" is shoved in there near the end, when the rest of the drop-down items generally follow a career path. In that path, "Student (High School)" should be first in the list, instead of what appears to be a momentary/singular attempt to alphabetize the list.
The Wikipedia entry for Subject-matter Expert, or SME, gives an appropriate general description, which is a similarly appropriate starting point in answering the question posed here: "a person who is an expert in a particular area or topic". As other people here have answered, the definition of SME in their own organizations can differ as to the specific qualifications that need to be held in order to be granted the "title" of SME, so let's turn to the US Office of Personnel Management's glossary definition of "Subject Matter Expert":
An individual within a department or agency who is thoroughly familiar
with a given topic and can provide expert input as to the appealing
aspects of a job and the skills and other qualities required of a job
In the US Government hiring in particular -- and in other large organizations that have many levels of positions and require specific expertise -- "subject matter expert" is a sort of level of position much like we often hear "entry level" or "junior" or "senior". And, much like those levels, there's room for interpretation. Someone who holds a PhD may be well on their way to being a SME, but so might a person who has worked their way through the ranks for years, regardless of degree, who is in some way known for their good work, often externally.
Again looking at the US Office of Personnel Management's words on the subject, but again recognizing that this information is applicable elsewhere, here is an example of how SME's are used within the organization for use in job analyses for other possible position openings (SMEs are also used as stakeholders in projects, and just about anything else that you could imagine someone with a lot of information and knowledge could be used for within an organization, besides their own actual tasks; SMEs are often called in as participants in other projects besides their own):
The term subject matter expert (SME) is properly applied to anyone who
has direct, up-to-date experience of a job and is familiar with all of
its tasks. The person might currently hold the job or supervise the
job. SMEs must provide accurate information and effectively
communicate their ideas. SMEs should rate the job tasks and
competencies for importance to successful job performance.
So, while in general use the term "Subject Matter Expert" can simply mean "the best person to judge specifics about something for which they have expertise" -- meaning, you could be the subject matter expert about cleaning the office refrigerator. In the context of labeling yourself a SME on a job application for the government or otherwise, it means you'd be easily recognizable as the top of the class, the cream of the crop, or another metaphor signifying that not only would you be able to walk in and be the expert on the topic (if that's what the job was looking for), you'd be able to judge the qualifications of others (if sitting on a hiring committee for which your domain expertise was valuable), and offer expert testimony, as Tanguerena mentions in this answer.