The holding period for a scalper may be measured in seconds. They're looking to make a living by scalping the spread, i.e. the difference between the bid and ask prices for a stock. To make such a slim-margin strategy successful you'd need volume and a ready market. This is not a strategy for the beginning investor; in fact it is not a strategy for the experienced investor either, unless you happen to be a floor trader on an exchange!
As the name suggests, day traders enter and exit their trades within the day, so that no open trades are carried overnight. Day traders use a variety of systems and rules to trade - scalpers are definitely day traders, but day traders can also use swing or momentum styles to identify their targets. They're basically looking for the same patterns as their longer-term brethren, but on a 1-minute or 5-minute chart rather than a daily or weekly chart.
If you look at a stock chart, you'll see that the price tends to move in brief waves, driven by the twin emotions of fear or greed. Swing trading seeks to take advantage of the timing of these short-term trading patterns to enter the market, The entry is usually in the opposite direction to the last price move, in anticipation of a reversal, so it's a short-term form of contrarian trading. Swing traders will normally have a set of 'rules' telling them when to enter and exit positions (for example Alexander Elder's Triple Screen strategy). The typical holding period of a swing trader would normally be around 2-5 days.
Momentum traders seek to jump on board a fast-moving stock, on the grounds that a stock, once moving, will tend to continue in the same direction (until it doesn't!). Momentum traders differ from swing traders in that swing traders will try to anticipate a change of direction, while momentum traders will wait for an established trend. Holding periods will vary. The idea is to exit when the trend slows down, so a trader looking at a 1-minute chart will clearly have a different view to a trader looking at a daily chart.
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