This is a long and nerdy post. I apologize.
Recent years have seen a blossoming of productivity applications, as well as the popular Getting Things Done approach, pioneered by David Allen.
At its heart, GTD is a method of keeping lists. It's not that different from what my mother-in-law did: she kept careful lists for her family camping trips, so she wouldn't forget something when they were 20 miles from a grocery store (think toilet paper).
What separates GTD is a greater clarity in your tasks (that is, you should create tasks that are clear and actionable), a series of different lists, projects, and contexts in which items are placed, and a careful review of all those tasks on a regular basis. (There's lots of information in Allen's book and on the internet, if you care to read up on GTD.)
With computers, GTD has moved to our various devices, and there's an astonishing number of very powerful applications out there to manage your lists. Although you can get by with pencil and paper, computers keep the lists cleaner, protected (they are backed up and in multiple places), and can automatically sort and hide tasks as needed. The dreadful bit of GTD before personal computers would be the endless re-copying of the lists. The dreadful bit of computerized tasks is the easy ability to fiddle and click, instead of actually getting things done (yes, this is highly ironic).
The debate on Macs is between Things and OmniFocus. Like a decent number of people, I was a Things user until just recently, when I switched to OmniFocus. Both are terrific programs, with pros and cons to each. Both have iOS applications that synchronize with your data. Both can kick to-dos down the road to a future date, which is one of the killer features of a GTD application; that is, you can tell the program, I don't want to see or think about this until 2 months from now, and instantly it is gone until the 2 months have elapsed. Both have quick entry panels for the Mac, so you can enter a task on the fly without switching into the application, and both can put projects on hold. On the cost side, neither application is particularly cheap, with Things being slightly cheaper (Remember the Milk is online, cute, and free, but lacks the power of these other apps).
Things
Things is easier to start with, prettier, and very user-friendly. One grasps its essentials fairly easily. It has a powerful system of tags, and a good system of keyboard shortcuts. The iOS apps work well, especially the iPhone app. The Today (that is, your highest priority) list is terrific, and you can quickly make things appear on the Today list, or take them off. Its downsides, however, become obvious with time: there is lots of flipping and scrolling through lists, and there is the need to synchronize your iOS devices on your local network via wifi (as well as wait for the sync to complete). Worse, after launching Things on your iPhone, you will have to wait for the sync to complete before doing anything, then enter whatever it is you wanted to enter, and then relaunch it to sync with your computer. This gets real old (they are working on a sync with their server, but it is taking forever to launch). The application is not updated very often, as you can see from their blog posts; the iPad version had a bug that lingered for months. This was one of the final reasons for my switching, as I tired of waiting for them to fix things or catch up to OmniFocus. The iPad app has page turn animations that get tiresome, and there is a lot of tabbing in and out of things on it and the iPhone; for example, you have to click on the task and then click edit to rename it, and then click again to save, instead of simply click and renaming it directly. If you include with a task an attachment or link to an email, you can't later delete or move that file or email, or the link is broken and unavailable to Things.
OmniFocus
OmniFocus is a workhouse, and powerful beyond belief. It has even better keyboard controls than Things, lots of flexibility in terms of its appearance and handling of your tasks, and a really powerful quick entry panel. It's easier and faster to schedule things into the future than Things is, and you can even schedule things according to a certain time (such as 2 pm). Projects can be dropped, which means the data is still in your system but it is gone from your lists. Projects can be made into templates, so they can be duplicated. You can have folders and containers, and tasks can become subtasks. The sync is terrific, as there is a free sync with their servers, so your data is synced, anytime anywhere (although the sync can be frustratingly slow, and sometimes it crashes). The iOS apps are great, with home buttons to get out of something you've tabbed deeply into, and the totally, completely amazing Forecast view. There is also a quick entry button so that you can enter a task on an iOS device while it is still syncing with the servers (although this does crash on me, from time to time), and even a map/GPS feature so that tasks can be keyed to GPS locations. The Omni Group is quick to update, fix, and innovate, which has led many people to abandon Things.
On the downside, OmniFocus is very granular and not easy to grasp at first, nor is it as visually appealing as Things (I have a strong feeling that 2.0 will change this, as the OmniFocus iPad app is gorgeous, and, like Apple, The Omnipresent Group will follow their iPad design leads). The Omni Group insists on using inspectors instead of a some sort of panel, and my inspector is always getting lost or in the way. Search the internet and you find people frustrated with OmniFocus' sharp learning curve. There's some early feelings of fear when you don't see the actions that you know are there (until you finally realize that you're in the wrong view, and those tasks are still in there). OmniFocus also forces you to put your actions into projects, which doesn't always make sense; if you're like me, you have lots of tasks that don't really belong in a project. Further, The Omni Group keeps their applications rather pricey, even in the days of the App Store and digital downloads that are driving prices down. Fortunately, there is a generous academic discount.
I must say that it was fairly scary to switch. I was surprised how nerve-wracking it was; that data is rather precious, and the fear of losing or missing something is surprisingly powerful. Going from a trusted system to a brand new one is stressful. After a week or so, however, I found myself relaxing and trusting OmniFocus.
Whatever you use, the goal is to be productive and stress-free, and to create as little friction between you and the plates you are currently spinning. Good luck.