A discretionary stock broking service is one that offers portfolio management as well as dealing. Clients can simply had over their money to the broker and the broker will make all investment decisions. Clients can impose constraints such as ethical investing requirements and retain some involvement in the process if they wish.
Brokers usually charge a percentage of the value of the portfolio annually for discretionary portfolio management. However, they do not usually charge commission for each trade made on a discretionary account. This means that brokers are not tempted to churn (trade more frequently than necessary) clients' investments in order to generate more commission. It does however mean there is some temptation for brokers to invest conservatively and then do little actual work on a discretionary portfolio.
Discretionary broking is usually the most expensive way of investing, and it offers the investor the lowest level of control over their investments. It does mean the least work for the investor and should mean the most work put into portfolio management by the broker. The amount of work required to provide a discretionary service means that most brokers are only willing to offer discretionary broking to clients who have portfolios of a reasonable size (typically £50,000+, often considerably more).