There are so many different options. I really haven't found anything that's exactly right for me. I've been trying to adapt GTD for personal use. I also need something that has a web version.
I am a Google Apps user and right now I use GQueues for organizing tasks. I really like it, enough to have purchased the full version (I don't remember what's missing from the free version), but it doesn't have a native Android app (although the developer is getting lots of pressure to create one). I use the HTML5 mobile web interface on my phone. It's not ideal for adding tasks from my phone but its other strengths are enough that I can live with it. One big plus for me is that GQueues (the paid version) allows me to sync with Google Calendar, which is how I get phone reminders (you can do notifications, email, and/or SMS, and I have a Chrome extension for desktop notifications). I can display any scheduled task on my Android home screen with the calendar widget. That works fine for me because if I haven't scheduled a task, I don't miss it from a home-screen display. But this doesn't allow you to check off completed tasks; you have to use the mobile web version for that. I really, really like the full web interface but it's a bit sluggish. If you like to structure your tasks in hierarchies, it's great because it doesn't limit the number of levels you can create. I've used it in the past for collaborative work projects, too.
Because I'd be interested in a purpose-designed widget, I'm currently test-driving a combination of ToodleDo (which I'm not liking) as a web back end and two different Android apps that interface with it: Ultimate To-Do List and DGT GTD. Both have home-screen widgets (for display and access to the app) and reminder capabilities. Ultimate To-Do list is powerful and can sync with Google Calendar but I think its interface is terrible. The free app is full-featured but only a 14-day trial, which just isn't long enough for me to decide. The only feature I've found that it has over and above the other is location–it will notify you if you have a task configured with a location that is in proximity to where you are. That's not useful enough for me to counterbalance the interface.
The interface of DGT GTD is better, not great, but the app is pretty flexible and it's growing on me. You can create several different kinds of structures for organizing your stuff: folders, projects, checklists, contexts. Hierarchies are limited, though, and I don't think it can do calendar syncing. The version I have is only an alpha; I would guess it will ultimately be a paid app. If I could live with a stand-alone app, I would seriously consider DGT GTD. The biggest problem I have with either of these ToodleDo interface apps (there are others I haven't tried) is that I am really picky when it comes to interfaces and theirs just don't do it for me (apps or widgets). I really don't like ToodleDo's interface, either, and I'd need to pay to get the features I need. I hate paying for any program whose interface I don't like.
I've tried a number of other Android apps. Wunderlist is pretty flexible. I liked it but when I tried it, it couldn't do recurring tasks, which is a deal-breaker for me. It's been a while since I tried it, though. Remember The Milk gets lots of good reviews; I just never liked it, especially its web interface. GTasks syncs with Google Tasks if you use that. Its widgets are a little better but Google Tasks is too limited for me and you have to pay to get recurring tasks in the app. Astrid gets good reviews. It also syncs with Google Tasks and has paid add-ons for location awareness and widgets. It's on my list to try if I continue experimenting. I'm a heavy Evernote user and ReQall has the (paid) ability to sync with it, so I might investigate that, as well.
For really light weight, unstructured purposes, I was just using Evernote. While it's still a mainstay for me, I recently discovered GNotes, which has a widget that I am addicted to. You can choose any note to display on your homescreen. I keep one for my shopping list and one that's kind of my catch-all notepad; anything that really needs to be organized on a continuing basis ends up in GQueues (tasks) or Evernote (documentation). I really like how Gnotes handles checklists. It also lets you set reminders and it's got some other nice features. It has secure notes (pattern or pin), which I've used to temporarily remember passwords, and it syncs with Gmail; all of your notes show up as archived email messages with the label "Notes." So far, I'm finding this latter feature quite useful. It can sync with a web version as well but the Gmail and web syncing are mutually exclusive. If it looks like it's going to develop into a stable app/service with solid features and decent support, I might consider using it more heavily, but it's hard to imagine transitioning away from Evernote.
I'd love to know about other combinations of web and Android apps that others have used.