The base is the historical form from which other forms derive.
For example, the historical base for "king" is malk. All forms can be explained from it. If you add a possessive suffix, it becomes מַלְכּוֹ "his king," מַלְכֵּ֫נוּ "our king," etc. The independent form has undergone modification from this base by the insertion of segol to relieve the problem of a final consonant cluster (malk → málek), and the stressed syllable was then "harmonized" (Kutz & Josberger, p. 76) in some way, it generally becoming a corresponding long vowel (qudš → qúdeš → qṓdeš), a segol (málek → méleḵ) or a pataḥ in the case of gutturals (which might also change the initial segol).
The historical base for segolate plurals, however, is bisyllabic, whereas the singular was monosyllabic. So, the base for "kings" is malak (rather than malk). This is how we derive the plural məlāḵîm with an initial reduced vowel. You cannot reduce a vowel that is caught in a closed syllable (the progression from malk to malkim to malakim to mlakim is impossible).
"Structure" refers to the set of vowels that are on a given root with its various affixes.