Symbolism, Imagery, Allegory
There sure are a lot of moms in this play, aren't there? You've got Eleanor, the mother of King John; you've got Lady Falconbridge, the mother of the Bastard; and you've got Constance, the mother of young Arthur, all of whom are strong and important characters.
John and Arthur are especially loyal to their mothers, who are the biggest supporters of their sons' competing claims to the throne. What's interesting (okay, bizarre) is that, at one point or another, each and every one of these moms is accused of being a floozy who cheated on her husband and gave birth to a "bastard." What's that all about?
Well, the biggest issue in this play is legitimacy: Arthur and John are both descendants of King Henry II, but who should have taken the throne after Richard I died? Since the throne is inherited by family members, political power depends completely on women sleeping only with their husbands. If there's any doubt about who the father is, then the child is left out in the cold.
As you can imagine, this led to a ton of paranoia, with people constantly worrying about women being unfaithful cheaters. The Bastard expresses this common cultural prejudice when he says that he knows who his mother is but has doubts about who his father is, "as all men's children may" (1.1.64).
Okay. That pretty much explains why Constance and Eleanor go around accusing each other of cheating on their husbands. If one could prove the other kid is illegitimate, then that kid wouldn't have the right to be king of England.
But it isn't just actual mothers that make their presence felt in this play: there are metaphorical mothers as well. You want us to give you an example, right? Right. When the Bastard is criticizing the English lords for rebelling against King John, he describes them as "You bloody Neroes, ripping up the womb / Of your dear mother England" (5.2.153-154). Basically, the Bastard is comparing his listeners to the Roman emperor Nero, who famously killed his own mother.
Okay, fine. What's the effect of all this? Well, when the Bastard uses this metaphor, it suggests that the English people are like one big family. It also kind of suggests that King John is like a parent to his subjects, who should be loyal, not rebellious. By the way, Shakespeare's own monarch Queen Elizabeth I went around referring to herself as a "mother" to her English subjects in order to encourage them to behave like loyal children (source).
But the problem with the Bastard's whole "be loyal to mother England and King John" metaphor is this: everybody knows that England is their "dear mother" (5.2.154)—or, at least the English characters do. But what they don't know is who their patriarchal "father" is. In other words, is John a legitimate king who had a right to take the throne when Richard I died?