For many years, the focus of education was geared right down the middle, so to speak; students who struggled and students who excelled were virtually ignored. Of course the pendulum has swung back and forth several times on both extremes over the years, and now both groups are given sufficient opportunity to succeed in most educational environments.
Students who display extraordinary abilities in some area can be considered gifted, and teachers can play a significant role in enhancing their academic experiences as well as their talents. For students whose specific gifts are music, art, acting, or some other form of the arts, teachers can be most helpful in several ways.
Teachers can encourage and appreciate these students' gifts. So many times we hear stories of adults who have succeeded despite their teachers; they got criticized or belittled by teachers at some point in their lives and they used that as a motivation to succeed. How much better it is to hear the stories of teachers who were inspirational because they believed in their students and supported their efforts. Teachers who have nothing specific to offer these students in terms of instruction (because the gifts are out of their fields of expertise) can still attend recitals, performances, and showings. They can encourage simply by acknowledging present talents and future dreams.
Obviously teachers who have something more concrete to offer should be giving these exceptional students opportunities to perfect and display their talents. Use them to help others in the same field so they can learn to articulate the mechanics of their talents but also sharpen their own skills by teaching. Help them find the right opportunity after graduation, whether that is college or some kind of specialized training, and use any contacts to provide guidance and/or opportunities to continue showcasing those talents.
For students with exceptional intellectual abilities, teachers must find ways to stimulate continues inquisitiveness. One of the most tragic (and common) things these students experience is boredom because the class material is below their intellectual capabilities. Teachers can offer opportunities for further study or for these gifted students to tutor or teach others. This serves both as a challenge to the gifted student and an effective reminder that not everyone has his capabilities (as well as benefiting the recipients of the instruction). This is a bit tricky, as students are quick to call out "teacher's pet" in this situation.
A less conspicuous but equally effective method of challenging exceptional students is to "up the play" for them in ordinary work. For example, when choosing a research paper topic, these students should never settle for the commonplace. Teachers should encourage them to go higher, deeper, or further in all work and refuse to let extraordinary students settle for the ordinary. The link below also has some helpful, specific suggestions.
Teaching is a high calling. Every student should be challenged to excel; those who achieve that goal early can be encouraged and motivated to improve by those charged with educating them: teachers.
My answer to this question will be a bit different. Having both of my children identified as gifted, I watched two completely different children handle the same way as described above. Their discontent was palpable. I decided for myself as a teacher that my gifted students would not do more work or work extended from what all the other students were doing, but truly different work. Sometimes it was the same material to learn such as becoming comfortable with writing complex sentences but was taught in a completely different manner. Examples of different work for my 8th graders were such things as writing Shakespearean insults within the guidelines provided, reading books with difficult vocabulary embedded in the book, debating such things as dropping the atomic bomb, right or wrong, or doing small amounts of teaching as a small group with one topic to the rest of the class which had not read the material presented. During the reading of Anne Frank, we experimented with whether my students could be silent if we played the card toss game Anne played with the others in the attic. We researched a site which had the students follow one person in the war, predict what would happen based on the information, and then check their results. I know that gifted students can often teach material to those who don't get it, but they get tired of that and want to pursue material that holds their interest. Students in that class had to do group work on the Harlem Renaissance with each person choosing a musician, an artist, a writer, and another category they chose. I had students arguing over who would get to do Billie Holiday and especially the song, "Strange Fruit." My methods didn't work with everyone, but many parents told me that their children would discuss this class at home with them, asking their opinion.