They usually input the trade on a work station. Sometimes, if the order is large enough (Usually 10000 shares) they will call a trader. A ticket is usually written out after these trades.
The best time to buy would be when a stock is trading below its intrinsic value.
Using different models (Discounted Cash Flow, PE-Earnings Growth,etc.) you can estimate the intrinsic value -- or...
To invest in the businesses success both to improve the businesses output and to receive a share of the profit. Some brokers buy stock purely to sell for profit.