Technical analysts identify areas of support and resistance in price charts. They are identified by observing levels that the price has consistently failed to break above or below. Most of the time the asset in question tends to trade between these two lines.
A breakout is simply when the price of an asset rises above an area of previous resistance, setting a new higher-high.
If the move is accompanied by a great deal of volume and eventually sets a low that is higher than the resistance line, this is considered to be a confirmation of the breakout and the beginning of a new trend.