A weak verb has one or more weak letters in it, often a guttural. See the charts in the back of KHW and this video. Some verbs have two such letters and are called doubly weak.
These verbs are classified by the location of the weak letter; e.g. a first-nun verb is a verb where the first letter is a nun. A third-guttural is a verb which has a guttural letter as its last letter.