Many pieces of classical music are divided into sections that are usually (but not always) marked by a pause in the performance, and these sections are often designated as movements. Such works can be symphonies, concerti, sonatas, and chamber pieces such as trios and quartets. However, it is also possible to use the term for the separate elements of suites, masses, sets of variations, and various kinds of program music. The divisions of operas and ballets are normally described as acts and scenes, thus reflecting their relationship with the theater rather than the concert hall.
Movements are frequently seen as having a degree of independence from the work to which they belong. Classical music radio stations play separate movements more often than full symphonies, etc, and many listeners are often unaware of how they relate to the complete work. There are indeed some movements that are so famous in their own right that they have virtually left home; examples include Widor's Toccata, which is actually the final movement of his Symphony for Organ No. 5, and the brilliant scherzo by Henry Litolff that is the 2nd movement of his Concerto Symphonique No. 4. In both cases, the vast majority of music listeners have not heard a single note, other than these movements, that was written by the composers in question.
It is not easy to generalize, but the standard symphonic (etc) first movement follows what is known as sonata form. That is, it has three sections, namely exposition, development and recapitulation. In other words, the theme or themes are presented at the outset, they are developed in various ways, and repeated in some form or other at the end. However, that statement is a huge over-simplification. For example, many movements include material at the beginning or end that falls outside this pattern, generally referred to as introductions and codas, and transitional passages that link the various parts together. There is also no rule that states that all the themes must be introduced at the beginning, or indeed how many themes there should be.
Another aspect of sonata form is the use of key structures within the movement. It is typical for the first and second themes to be in different keys, with changes from major to minor, or vice versa, modulated via a transitional passage. It is also common for the development to start in the same key as the exposition ended, and for the recapitulation to return to the keys used at the start of the exposition. However, great composers are adept at bending the rules to achieve their effects, and it is their originality in these and other matters that makes them great.
Whereas first movements are usually fairly brisk in pace (allegro), second movements are often much slower (adagio or andante), and sonata form is not expected. Second movements are commonly referred to as slow movements, although this is a relative term, and the slow movement does not always come second. A good slow movement can indeed be "moving" and emotional, sometimes comprising a single long tune that develops more from subtle key changes than from the introduction of secondary themes. A famous example of this would be the adagietto that comprises the 4th movement of Mahler's 5th Symphony (which has five movements), made famous as the theme of Visconti's film "Death in Venice" (1971). There is no rule that states that a second or subsequent movement must relate directly to the opening movement, but this is often the case.
Symphonic structure has changed over the centuries, such that from the 18th century it became common practice for symphonies, sonatas and quartets (etc) to comprise four movements, whereas concerti only had three. The "extra" third movement was typically a minuet and trio or a scherzo. A minuet is basically a dance in triple time, and a trio has a simple three-part structure of two contrasting sections with the third part being a repetition of the first. The word "scherzo" literally means "joke", and is generally a relatively playful and lighthearted section, in the nature of a fast minuet. It is also quite common for the scherzo to lead straight into the fourth movement without a break. For example, many listeners think that Beethoven's 5th Symphony is a three-movement work for this reason, whereas it does actually have four movements.
Final movements are the opportunity for the composer to pull everything together and build towards a climax that will inspire the audience to break into loud and prolonged applause. That at least has been the pattern since the Romantic era, and there is plenty of choice for the concert planner who wants to end the evening on a high. Sonata form is common for final movements, as are long and complex codas that allow the performers a final flourish. In concerti, the soloist can give it their all to earn their bouquet!
One special feature of movements in concerti is the cadenza. This is a passage in which the soloist (or soloists in double concerti, etc) plays on their own with the orchestra silent and the conductor at rest. In earlier times, it was usual for the soloist to improvise at these points, and this often gave rise to problems when they would soar off on flights of fancy of their own, sometimes having great difficulty in returning to where they started from. This practice has faded since the Classical period ended, with most cadenzas being written by the composer, although the soloist still has the opportunity to show his/her individualism in how they interpret the tempo of the piece, etc. It is noticeable that some modern performers have re-invented the improvised cadenza in performances of works by, for example, Vivaldi. Nigel Kennedy's highly acclaimed interpretation of the Four Seasons is a case in point.
As stated above, it is not easy to be hard-and-fast when describing what movements look and sound like. There are so many variations on the theme that generalizations are bound to be accompanied by hosts of exceptions.