A fleeting vowel is a vowel that appears in the stem of a noun in one case form but in none of the others. For masculine first declension nouns, that vowel appears in the nominative singular(†) but in none of the other case forms. For other noun types that vowel appears in the genitive plural(‡) but in none of the other case forms. The vast majority of fleeting vowels occur in words that have suffixes containing к or ц; the most common patterns of this type are shown in these reference pages. Some fleeting vowels occur in foreign borrowings that coincidentally have к or ц that are unrelated to the Russian suffixes (nominative singular маска becomes genitive plural масок). Some fleeting vowels occur in native Russian words without those suffixes (пёс, лоб); these just have to be memorized as the student encounters them.
Pay particular attention anytime you are dealing with the genitive plural of a noun that would normally end in "а" or "о" in the nominative singular. If the vowel drops out in the genitive plural, leaving the word with a consonant cluster at the end of the word, there is a good chance that a fleeting vowel will appear in the genitive plural to break up the consonant cluster. For instance nominative singular девушка becomes genitive plural девушек, and nominative singular кресло becomes genitive plural кресел. Double check the genitive plural with a dictionary the first time you encounter such a word.
† and the accusative singular when it copies the nominative singular
‡ and the accusative plural when it copies the genitive plural