Palatalization generally refers to two phenomena:
As a process or the result of a process, the effect that front vowels and the palatal approximant [j] frequently have on consonants;
As a phonetic description, the secondary articulation of consonants by which the body of the tongue is raised toward the hard palate during the articulation of the consonant. Such consonants are phonetically palatalized, and in the International Phonetic Alphabet they are indicated by a superscript 'j', as [tʲ] for a palatalized [t].
The second may be the result of the first, but they often differ. That is, a vowel may "palatalize" a consonant (sense 1), but the result might not be a palatalized consonant in the phonetic sense (sense 2).
Conversely, the word palatalization may also be used for the effect a palatal or palatalized consonant exerts on nearby sounds, as in Erzya, where the front vowel [æ] only occurs as an allophone of [a] after a palatalized consonant, as seen in the pronunciation of the name of the language itself, [erzʲæ]. However, while the process may be called palatalization, the resulting vowel [æ] is not called a palatalized vowel in the phonetic sense. Terminology such as "palatal vowel" is found, however, but this is primary and not secondary articulation.
Phonological description
"Pure" palatalization is denoted by a small superscript [ʲ] in IPA. This is a modification to the articulation of a consonant, where the middle of the tongue is raised, and nothing else. It may produce a laminal articulation of otherwise apical consonants such as /t/ and /s/. It is a phonemic feature in some languages; a common misconception is that it's merely allophonic, like in English. Phonemic palatalization is contrasted with either plain or velarized articulation. In Baltic-Finnic languages, the contrast is with plain consonants, but in Russian, it is with velarized consonants.
Phonetically palatalized consonants may vary in their exact realization. Some, but not all languages add offglides or onglides. In Russian, both simply palatalized consonants, and consonants with a long, noticeable palatal offglide are found. Furthermore, /tʲ/ and /dʲ/ have a fricative release. For example, пальто [palʲto], царь [tsarʲ], Катя [katʲa], respectively. In Hupa, on the other hand, the palatalization permeates the consonant, and is heard as both an onglide and an offglide: [aʲkʲa].
Palatalization is not the same as primary palatal articulation. These contrast in Skolt Sami; e.g. plain (or velarized) alveolar nasal <n>, palatalized alveolar nasal <'n> and palatal nasal <nj>.
Phonetic (synchronic) palatalization
Palatalization may be a synchronic phonological process, i.e. some phonemes are palatalized in certain contexts, typically before front vowels or especially high front vowels, and remain non-palatalized elsewhere. This is usually phonetic palatalization, as described above, but need not to be. It is usually allophony, i.e. it may even go unnoticed by native speakers. As an example, compare the /k/ of English key with the /k/ of coo, or the /t/ of tea with the /t/ of took. The first word of each pair is palatalized, but few English speakers would perceive them as distinct.
The variation might be seen as allophonic variation as long as the "palatal" sound causing the palatalization is there. However, syncope or elision might delete this sound, and thus only the palatalization remains as a distinct feature. This process is widespread in Baltic-Finnic languages, which have lost their original (Uralic) phonemic palatalization but some have regained it from Slavic. For a minimal pair, consider Estonian kass [kasʲ:] from *kassi "cat" vs. kas [kas:] (interrogative).
Sometimes palatalization is part of a synchronic grammatical process, such as palatalizing the first consonant of a verb root to signal the past tense. This type of palatalization is phonemic, and is recognized by the speakers as a contrasting feature. However, what may have started off as phonetic palatalization can quickly evolve into something else, so not all of the resulting consonants are necessarily palatalized phonetically.