Morphemes can be defined as the smallest grammatical unit of meaning and sound. (This is debatable, since there is much theoretical evidence for silent morphemes (such as the inferred "you" in English imperatives-- "(you) do this!"), as well as sound sequences recognized as 'morphemes' but which seem to have no meaning.)
There are several different types of shapes of morphemes (or morphological processes).
The stem is often the same as the root, and thought of as the part of the word with the most meaning, and it is the base off of which words are built. Some examples of simple words which are only stems are "cat," "run," and "red." Any of these can undergo morphological processes to indicate various different meanings, including number (cat/cats), tense (run/ran), a shift in syntactic function (cat/catty, run/runner), or subtler nuances (red/reddish).
The simplest and most common type of morpheme (besides the root), affixes are easily-separable word components. Most common are suffixes, which follow the root (such as the English plural -s/ -es), and prefixes, which come before the root (e.g. "un-", "de-", "re-".) Much rarer are infixes, inside the root (English "expletive insertion" e.g. "abso-bloody-lutely") and circumfixes, around the root (German ge- -t, e.g. the verb spielen has the past participle gespielt).
In some languages different types of affixes will preform separate functions; for instance, in English, suffixes are more likely to impart grammatical meaning, while prefixes are more likely to cause semantic meaning differences.
Root modification, also known as apophony, involves making changes to the root itself. A specific form of this process called vowel gradation or ablaut is common in Germanic languages, which involves changing the vowels in a stem; it survives in English irregular (or "strong") words such as run/ran, sing/sang/sung, or goose/geese, mouse/mice. These largely arise from historic vowel harmony caused by various affixes.
The Semitic languages extensively employ root modification (known as root-and-pattern morphology or Semitic trilateral/ consonantal roots) to convey a broad variety of linguistic meaning. For instance, the Arabic root KTB can be coupled with various vowels in forms such as kitaab "book", kitab "to write," kataba "he wrote," or kaatib "writer."