Intuition for irregular languages
The language $L_1$ is the quintessential example of a non-regular language. Intuitively, finite automata cannot count, so they cannot ascertain that the number of $a$'s is the same as the number of $b$'s. Alternatively, an automaton must &quot;remember&quot; how many $a$'s it has seen, which is impossible since there are infinitely many possible number of $a$'s. This intuition can be formalized using Myhill–Nerode theory: if $q_n$ is the state that a DFA for $L_1$ is at after reading $a^n$, then $q_n \neq q_m$ for $n \neq m$, since $a^nb^n \in L_1$ while $a^mb^n \notin L_1$; so there must be infinitely many states.
The language $L_2$ corresponds to the regular expression $a^*b^*$, so is regular.
The language $L_3$ is a &quot;generalization&quot; of $L_1$ (this can be made precise: $L_1$ is a quotient of $L_3$, $L_1 = L_3/\{cd\}$). Once again, an automaton needs to count how many $a$'s there are, which is impossible.