For all practical purposes, a pip is to a Forex trader what a penny is to stock traders. It's the main point of price movement tracked by traders in discussions of price movement and calculations of trade sizes for risk management. (And yes, this is still the case even if your broker quotes an extra digit or two after it. More on that below.)
Rule of thumb: Take the lowest denomination that a country's bank accounts would denominate its local currency in and add two more decimal places.
For the majority of currency pairs, one pip is always 1/100th (one hundredth), or two additional decimal places, of the lowest physical denomination of the quote currency (the currency on the right side). Since most currencies have an equivalent to a cent or penny (ie. two decimal points is the lowest denomination in bank accounts, with one cent/penny being .01 of a dollar), the pip value would be one hundredth of 0.01 which is 0.0001 - the 4th decimal.
One of the most prominent exceptions is the Japanese Yen (JPY). It has no cent/penny equivalent even within Japan. One Yen is essentially the local Japanese equivalent to one penny even within its domestic use (a 3 million Yen salary is economically and roughly class equivalent to a 30k USD salary and people buy 1500 Yen DVDs without any more hesitation than an American would buy a USD $15 DVD.) So considering its lowest denomination is 1 JPY, then the pip value of any Forex pair quoted in JPY is the 2nd decimal (since two additional decimal places makes 0.01 of a Yen).
If the price of EUR/USD (Euro vs US Dollar) is 1.3602, then the "2" is the current pip value (if it changes from 1.3602 to 1.3603, we say that it has gone up 1 pip). The USD, or US Dollars, is the quote currency (right side) in this case which has a lowest denomination of a cent (0.01 of a dollar) so the pip value at two additional decimal places is the fourth.
If USD/JPY (US Dollar vs Japanese Yen) is priced at 89.05, then the "5" is the pip value (if it changes from 89.05 to 89.06, we say that is has gone up 1 pip). The JPY is the quote currency (right side) in this case, and its lowest physical denomination in cash is one Yen, so the pip value at two additional decimal places is the second.
So why do brokers tend to quote one extra digit now?
In recent years, many brokers are quoting an additional digit, resulting in a 5th decimal for most currency pairs (or a 3rd decimal in the case of JPY-quoted pairs.) These 5th/3rd decimals are commonly referred to as "fractional pips" or sometimes "pipettes" since they are one digit past the pip value (one tenth of a pip.) The actual price movement on the market is still based around the actual pip values (4th for most/2nd for JPY-quoted pairs), so the fractional pip values are insignificant in terms of trading decisions and every-day trader talk. In fact, it's normal for a trader to discuss the fact that a currency pair moved 105 pips today. It'd be incredibly odd for anyone to discuss a 1050 pipette/fractional pip move.
The fractional pip trend came around the same time the stock markets began offering fractional pennies - it's not rare to see equity ECNs fill a stock at say 10.055 even if the bid is 10.05 and ask is 10.06, for instance.