Secondary stress (or obsolete secondary accent) is the weaker of two degrees of stress in the pronunciation of a word; the stronger degree of stress is called 'primary'. The International Phonetic Alphabet symbol for secondary stress is a short vertical line preceding and at the foot of the stressed syllable: the nun in proˌnunciˈation. Another tradition in English is to assign acute and grave accents for primary and secondary stress: pronùnciátion.
Most languages, if they have stress at all, have only one degree of it on the phonemic level. That is, each syllable has stress or it does not. Many languages have rhythmic stress; location of the stress may not be predictable, but once the location of one stressed syllable (which may be the primary stress) is known, certain syllables before or after can be predicted to also be stressed; these may have secondary stress. An example is Dutch, where the rule is that initial and final syllables (word boundaries) take secondary stress, then every alternate syllable before and after the primary stress, as long as two stressed syllables are not adjacent and stress does not fall on /ə/ (there are however some exceptions to this rule). See Dutch phonology: Stress. A similar rule applies in Romanian: secondary stress falls on every alternate syllable, starting with the first, as long as it does not fall adjacent to the primary stress. In other languages (including Egyptian Radio Arabic, Bhojpuri, Cayuga, Estonian, Hawaiian, Kaure, Malayalam, and Warrgamay), secondary stress can be predicted to fall on heavy syllables.
However, in other languages the placement of secondary stress is not predictable, or may not be predictable (and thus be phonemic) for some words. This is frequently posited for Germanic languages, including English. For example, secondary stress is said to arise in compound words like vacuum cleaner, where the first syllable of vacuum has primary stress, while the first syllable of cleaner is usually said to have secondary stress. However, this analysis is problematic; Bolinger (1989) notes that these may be cases of full vs reduced unstressed vowels being interpreted as secondary stress vs unstressed. See Stress and vowel reduction in English for details.
In Norwegian, the pitch accent is lost from one of the roots in a compound word, but the erstwhile tonic syllable retains the full length (long vowel or geminate consonant) of a stressed syllable; this has sometimes been characterized as secondary stress.
See also 
- Ioana Chițoran (2002) The phonology of Romanian, p 88
- Gjert Kristoffersen (2007) The Phonology of Norwegian, p 184