Event A: The coin comes up heads.
Event B: The coin comes up either heads or tails.
The probability of A is $\frac12$; the probability of B is $1$.
For less trivial examples you need more than two possible outcomes. Flipping a coin twice (or flipping two coins) will work, as will rolling a die. With two coin tosses, for instance:
Event A: I get at least one head.
Event B: I get at least one tail.
These are not mutually exclusive: both occur if I get HT or TH. They’re also not identical: if I get HH, A occurs but B doesn’t. Each has probability $\frac34$.
With a die:
Event A: the number that comes up is even.
Event B: The number that comes up is $1,2$, or $3$.
If I roll a $2$, both A and B occur, so they’re not mutually exclusive. Note that in this case each has probability $\frac12$, so their probabilities do add up to $1$, even though they are not mutually exclusive.