When iPhone 4S was released, it introduced Apple fans to a concept most mobile users had known for quite some time: voice actions. The general reaction to Siri ranged from "Android did that already" to "I don't see how this is actually useful." While it's difficult to quantify what is useful for every user, Siri most certainly is not a clone of any previously available voice command program.
The big difference is in how it parses commands. Instead of being forced to use specific canned commands to accomplish a task, Siri is able to understand speech contextually. When I say contextual, I mean the difference between saying "Text John, message: I'll be late for lunch" vs. "Tell John I'll be late for lunch." You can talk to it naturally more or less. Siri strives to be conversational, often asking you for clarification when you haven't given it enough information to process your query. While it doesn't exactly operate on magic and wonder as Apple's marketing team would have you believe, it is a pretty big leap ahead of what we've seen previously on mobile devices. Not having to speak in a certain syntax removes one of the big obstructions to the adoption of voice activated computing.
Where Siri is most useful is cutting down on the steps you need to achieve a given action. For example, changing settings, or setting alarms and timers. You COULD navigate through the menus to get to whatever setting you like, but that takes longer than simply saying "set timer for 15 minutes." The key is: using Siri doesn't take you out of whatever app you're in. The menu pops up when needed, and goes away once it's finished, never breaking the flow of what you’re working on.
Let's say you get an email that contains a meeting request or an appointment. Instead of going back out to the home screen, finding and opening the calendar app and inputting the information manually, you can activate Siri while still in the email app and dictate your appointment details there. Once you're finished, the Siri prompt goes away and you're right back to work. Those minutes and seconds saved add up over time. Not having to exit the app also keeps you focused on getting things done.
Use it to dictate tweets or notes. Use it to check your email or read your notifications. Use it to check stats or math problems. Use it for almost anything! After using iPad mini for the last several weeks, using an iPhone 4 (which does not have Siri) is a bit painful. Once you get in the habit, it's easy to see why this feature was so coveted, and why it's so hard to go back to a phone that doesn't have it
Here's a fairly comprehensive list of the things you can do with Siri. Try them out and see how they work for you.