Going Long
To go long on a currency you are betting that the base currency will increase in value relative to the quote currency. You would therefore be buying the base currency and selling the quote currency, expecting the former to appreciate against the latter, thus realising a profit upon closing the trade.
Going Short
Going short on a currency pair, e.g. EUR/USD, involves speculating that the base currency (EUR) will decrease in value relative to the quote currency (USD).
When you go short on a currency pair, you are selling the base currency and buying the quote currency with the expectation that the base currency will decrease in value.
For example if you short the EUR/USD pair, then at the point of entering the trade you are selling Euros and buying Dollars. If upon closing the trade, the Euro has weakened against the Dollar, then a profit will be made. This is because the Euros can be repurchased at a lower price then they initially sold for, thus making a profit on the difference.