Uninfected words
There is no such thing in linguistics, simple as that.
It's really easy to confuse it with "uninflected" (which is what I think did happen), but if this is indeed what you saw on Wiktionary, it was a typo, a machine translation error or something like this.
Uninflected words
From Wikipedia:
In linguistic morphology, an uninflected word is a word that has no morphological markers (inflection) such as affixes, ablaut, consonant gradation, etc., indicating declension or conjugation. If a word has an uninflected form, this is usually the form used as the lemma for the word.
A lemma, or a base form, is the form of the word used to list this word in dictionaries and to generally refer to the word as a set of its all possible forms.
In Russian lexicography, the singular nominative is used as a lemma for nouns, the singular nominative masculine full for adjectives, the infinitive for verbs; and these are what sometimes is called the "uninflected forms".
It doesn't strictly conform to the definition above, because the sg. nom. of, say, the word река does have a non-zero flexion, whereas the pl. gen. form рек doesn't.
Most probably, one of the words you came across on Wiktionary had been a form of the word which happened to be a homonym for the base form of another word.
This could have been something like лис (pl. gen. of лиса "female fox") which is a homonym of лис "male fox"; Петь (neo-vocative of Петя, a male name) and петь "to sing"; пал (past masculine singular of пасть "to fall") and пал "controlled forest fire"; and so on.
To answer your questions:
Do one word, a small number of words, or a large number of words have them?
In the sense defined above, all words have them.
Some words don't have other forms, in which case they are usually called "invariable". There is a number of invariable words in Russian: function words like prepositions and conjunctions, adverbs, adjectival adverbs (деепричастия); also some borrowed nouns (метро, пальто, радио and so on). The percentage of invariable nouns in Russian is not big, although not tiny either, and some of them are used on the daily basis.
When should I use them?
You use them wherever you need to use this particular form: singular nominative for nouns, and so on.