What does it mean by "odds that an event will occur is $2$ to $3$"? 
In a swimming race, the odds that $A$ will win are 2 to 3 and the odds that $B$ will win are 1 to 4 Find the probability $p$ and the odds that $A$ or $B$ wins the race.

What does it mean by odds that an event will occur is $2$ to $3$?
 A: The odds are $2:3$ when the probability is $2/(2+3) = 2/5$.
Edit in response to comment.
Indeed odds are just ratios. $2:3$ is the same as $4:6$. I don't know why they came to be written that way, but I'm glad they are. If you think of them as fractions it's all too easy to confuse them with probabilities. Of course they can't be, since odds of $3:2$ make perfect sense but a probability of $3/2$ doesn't. 
Odds are the usual way to express stakes in a bet. If the probability of an event is $2/5$ (as in your example) then it would be fair to stake $\$3$ against $\$2$ in a bet. You and your opponent each put the stake ($\$2$ or $\$3$) into the pot; the winner takes it all.
A: The odds for an event are $x$ to $y$ if the probability of that event is $x/(x+y)$.
A: The intuition behind the notation $x:y$ is that there are $x$ unfavorable outcomes for every $y$ favorable outcomes. For example, the odds against spinning a black in European roulette are $19:18$, since there are $19$ red/green sections and $18$ black sections.
In other words, while a probability measures the ratio $\frac{\text{good}}{\text{total}}$, the odds against measure the ratio $\frac{\text{bad}}{\text{good}}$.
I do not know the history of this notation, but there is a reason for why we care about this ratio. If an event has odds of $x:y$ against, then it would be a fair bet to receive $x$ dollars if the event succeeds and $y$ dollars if the event fails. This is equivalent to receiving $x/y$ dollars for a success and losing $1$ dollar for a failure. That is, the odds against is the ratio of money won to money lost for a fair bet on that event.
