The short answer
Implied probability is the chance of an outcome that a given price corresponds to. For decimal odds the formula is one divided by the odds.
Odds of 2.00 imply 1 / 2.00, which is 0.50, or 50 per cent. Odds of 4.00 imply 25 per cent. Odds of 1.25 imply 80 per cent. That is the whole calculation.
Converting a price to a percentage is what lets you compare two things that are otherwise hard to compare: what the market is charging, and what you believe.
Why the numbers add up to more than 100 per cent
Take a two way market priced at 1.90 and 1.90. The first implies 52.63 per cent, the second implies 52.63 per cent, and together they imply 105.26 per cent.
Probabilities of all possible outcomes should sum to 100 per cent. The extra 5.26 percentage points is not a mistake. It is the bookmaker's margin, built into both prices, and it is how the business makes money on a balanced book.
That margin goes by several names. Overround, vig, juice and hold all point at the same thing. The important consequence is that raw implied probabilities are always inflated, so they are not a fair estimate of anything until the margin is removed.
A worked example
Suppose a tennis match is priced at 1.55 and 2.55.
The first price implies 1 / 1.55, which is 64.5 per cent. The second implies 1 / 2.55, which is 39.2 per cent. Together that is 103.7 per cent, so the margin on this market is 3.7 percentage points.
Now suppose another book prices the same match at 1.62 and 2.45. The favourite there implies 61.7 per cent. Same match, same two players, and two books disagreeing by nearly three percentage points about how likely the favourite is to win. That disagreement is the raw material for everything a scanner does.
What implied probability does not tell you
It is not the true probability of the outcome. It is the probability implied by a price that has a margin built into it and that reflects one bookmaker's opinion plus whatever money has been bet.
Removing the margin gets you closer to what the market actually thinks. That process is called devigging and it deserves its own explanation.
Even a devigged price is still the market's opinion rather than the truth. It is usually a good opinion, which is precisely why beating it consistently is difficult.
How to use it in practice
Convert before you compare. Two prices in different formats, or two prices with different margins baked in, are not directly comparable until both are percentages.
Use it to sanity check your own view. If you think something is a 60 per cent chance and the market implies 42 per cent, the useful question is not whether you back it, but why you disagree with a market this strongly and whether the answer is convincing.
The odds converter on this site does the arithmetic across decimal, American, fractional and implied formats if you would rather not do it by hand.