In forex trading we have pairs of currencies. Lets take the EUR/USD pair. Suppose that its price is 1.2550. This means that someone can buy one euro by giving 1.2550 dollars. So we are estimating the value of one euro in terms of dollars. In this case the EUR is the base currency and the USD is the quote currency in which the euro is being quoted. In USD/JPY pair USD is the base currency and JPY is the quote currency. 

If the price of the EUR/USD pair goes from 1.2550 to 1.2590 then the euro has appreciated relatively to the dollar because in order to buy one euro someone must give more dollars than before. So the euro is more expensive now in terms of dollars or you can buy more dollars by paying one euro. If the price goes from 1.2550 to 1.2500 then the euro has depreciated relatively to the dollar because now you can buy a euro by giving less dollars than before or by paying one euro you can buy less dollars. When one currency is appreciated the other one of the pair is simultaneously depreciating and vice versa.