If storage is an issue, you could free some up. For example, if you've lots of music or photos on your device, select the relevant tab, uncheck the sync box and then sync your device. The relevant media is removed. You can then perform updates and manage your apps, perhaps free up more space, and later restore your media by resyncing it.
Occasionally, you might find the 'other' section becomes massive. In our experience, this is usually down to you having a lot of in-app data (see Step 3) or failed app updates, which can happen on trying to update without enough free space. Resyncing should help; if not, a restore from a local backup.
2. Discover app sizes
Apps can be massive. Sizes are shown in iTunes and the App Store, but that's the size of the compressed download. Once installed, an app's size can balloon.
In iTunes, check app sizes by clicking on the Apps tab and selecting Sort by Size from the pop-up menu at the top of the apps list. Peruse the list, and if there are apps or games you no longer use, consider deleting them. You can do so by clicking Remove in iTunes; when you've done so for all apps you'd like to delete, click Sync.
Alternatively, tap-hold an app on your device to make all the icons jiggle and, for each, tap the cross icon and then 'Delete' to remove it.
3. Examine app data
Open the Settings app on your device and in the General category, select Usage. You see available and used storage and a list of apps. These are ordered by the total amount of space they require, including app data.
If you've several such apps taking up loads of room you need, consider deleting data. For example, if you subscribe to magazines, delete old issues from within each app. You can usually redownload issues later if you need to. If you fancy taking a speedy option and don't have a capped broadband connection, deleting a Newsstand app takes all its data with it. You can then download a fresh copy from the App Store and the latest issue.
Magazine and book apps aren't the only storage culprits, note. Dropbox can (optionally) store documents locally (by flagging them as favourites) and some video apps have download capability, so check those too.
4. Back-up app content
Should you no longer use a game or creative app, but think you might one day return to it, download its data to your Mac using the free version of iExplorer (macroplant.com/iexplorer).
Connect your device to your Mac via USB, select Apps from iExplorer's sidebar and select the app in question. Select the Documents and Library folders, Ctrl-click and select Export to Folder. (Alternatively drag them to a Finder folder.)
The contents of these folders can later be sideloaded into a fresh install of the app, meaning you won't lose your progress in a game that doesn't support iCloud, or could get saved compositions from a music-app back to your device with a minimum of fuss.
5. Use last-chance folders
If you tend to frequently download new apps, chances are some fall out of favour, but you might not necessarily know which. Create date-based folders (07-2013, say) and place apps within that you don't think you use any more. If you find yourself using one, 'rescue' it from the folder. Otherwise, delete the folder's contents after a few months, first backing up app data as necessary.
(Note: if you don't download apps to iTunes on your Mac, sync with it before deleting the apps, so you've a back-up you can later install to your device. You can of course redownload apps from the App Store, but only if they are still made available to you.)
This is a fairly ruthless app-management method, but it's useful for keeping installs current and ensuring you have space.