It's easy to see why some people would worry about coffee. First of all, because caffeine is a stimulant, it can cause jitters and insomnia. It can boost heart rate temporarily, which is why people with certain heart problems are sometimes advised to avoid it. Coffee, regular or decaf, can also cause stomach upset and heartburn.
Moreover, the effects of coffee and caffeine can vary from person to person, depending on genetic and other factors. For instance, caffeine's transient effect on blood pressure and heart rate depends largely on whether you are used to caffeine or not. For most people, any such boost isn't a danger. However, in people with multiple cardiac risk factors who rarely consume it, caffeine may increase the risk of a heart attack during the hour or two after drinking coffee.
The stimulant effects vary as well. In fact, a 2010 British study suggested that caffeine isn't stimulating for people who drink coffee regularly. For them, caffeinated beverages merely counter the effects of caffeine withdrawal (such as headaches and a decreased ability to pay attention) and only help restore or maintain a normal state of alertness. In contrast, infrequent coffee drinkers get more of a stimulant effect from the caffeine. The more caffeine you consume regularly, the more you build up a tolerance for it.
Coffee's effects also depend on the amount of caffeine and other compounds in it. This, in turn, depends on the type of bean, how the beans were processed, and how the beverage is prepared. For instance, it has long been known that large quantities of unfiltered coffee—notably espresso or that made in a French press, regular or decaf—can boost blood cholesterol. The likely culprits are cafestol and kahweol, which are diterpenes in the coffee's oil that are trapped in paper filters. (Interestingly, though, these compounds have anti-cancer properties.) If you have undesirable cholesterol levels and regularly drink large quantities of unfiltered coffee, try cutting down or switching to filtered coffee. Single-serving coffee pods, by the way, contain a filter.
Note for pregnant women: Some research suggests that high doses of caffeine raise the risk of miscarriage and birth defects. While the evidence for this is not consistent, to be safe, pregnant women should drink no more than one or two cups a day.