In fixed-do solfège, each note of the scale, starting with C, is given a set name: do, re, mi, fa, sol, la and si. Students are given exercises in which notes are positioned on various clefs and in various keys, and they are asked to read out the notes in order, using their solfège names. Exercises start very simply and become increasingly more complex, with a change of clef every note or large leaps between notes. In mixed-do solfège, the tonic note of the scale is called do, and every other note is named in relation to the tonic. For example, in D major, D would be do, E would be re and so on. Exercises in movable do are similar to those in fixed do, with the exception that when a melody modulates in movable do, the new tonic is called do and the names given to notes therefore shift to another set of pitches. Movable-do solfège works with diatonic music only; for atonal music, a fixed-do approach provides a more consistent approach.
In music dictation, the object of the exercise is to correctly transcribe what someone plays. This is usually done with an instructor at the piano. The instructor may tell the students ahead of time what the starting note or chord of the dictation is, or he may let the students figure that out themselves. Dictations can be as simple as one-voice melodies that stay in the same key the whole time, or they can be as complex as multi-voice, fully-harmonized examples that include modulations or atonal elements. Dictation can also be given for purely rhythmic passages as well.
Most rhythmic exercises are designed to teach students to correctly read rhythms that they will find in music. Exercises can be as simple as a student beating a steady quarter-note beat while vocalizing a rhythmic line on top if it, or they may require the student to do complex tasks like beating out multiple meters while vocalizing.
Sight-singing combines elements of solfège and rhythmic training, and it usually consists of asking a student to sing a new melody she has never heard or seen before by looking at the written music. Melodies for sight-singing, like other ear-training exercises, can range from the very simple to the very complex, including modulations, changes of meter, atonal elements and other advanced elements.
Working with a tutor is one of the best ways to improve your musical ear. Most college and university schools of music have ear-training tutors who work with individual students on all of these exercises.