Sync: what exactly does it do?

You can sync only to one computer or iTunes library at a time. If you change your computer or have to set up your computer HD again, all content on your device will be replaced with the content of the new iTunes library.
This has always been this way due to the Digitial Rights Management Apple has to keep to.

If your apps are grayed out in iTunes, check if your computer is authorized, About iTunes Store authorization and deauthorization, or if there are restrictions for app installation set on your device in Settings/General/Restrictions.

Every time you sync your device, iTunes will copy all media, that are not part of the library on the computer, from your device to your computer. If you selected to copy new apps to your iPad without asking, this will also work the other way around.

Check these articles about syncing as well:

iOS: Syncing with iTunes

Copying iTunes Store purchases from your iPod or iPhone to a computer

But what does synch do? I have an iPad and when I connect it to my MacBook Pro and try to update the OS, I get a message that says there are apps on the iPad that are not on the computer and might be lost if I update the OS. Does "synch" put the most recent version of everything onto the other device? Or does it just put what's on the computer onto the iPad? The messages or alerts I see don't seem inherently how I would interpret the word "Synch". Thanks

@jn1234

There's a difference between syncing and updating your iOS:

Syncing makes sure that the content on your computer and the iDevice stays the same or mirrors the selection you took in the settings in iTunes for music, movies etc.

Updating your iOS will replace the firmware of the device with a newer version. Therefor your device has to be erased during the process. The content will be copied back again when you use a backup to restore it after the installation.

If you have apps or other media on your iPad, that are not part of the iTunes library on your computer, you will be reminded to transfer them. Usually the sync process takes care of that automatically, but, since you're trying to update the iOS, there is no sync process prior to the installation. You'll have to copy them manually by right clicking on your iPad in the device list and choosing "transfer purchases". Do the same thing to back up manually choosing "backup" after that. The backup contains the app data and settings. You will need a working backup after the iOS update, to be able to get all your data back. More info on backups here: iTunes: About iOS backups

To make sure your bookmarks sync from your Mac to your iphone, connect your phone, click on the info pane, go down to the Advanced Section and select "replace bookmarks" on next sync.

Syncing makes sure that the info on different devices stays the same, but this is one option to get new or correct info to a device containing false info.

Thank you, that helps. Here's my frustration with the terminology Apple uses. "Synch" doesn't synch. In other words, to transfer content that I purchased on my iPad from my iPad to the MacBook, I have to take steps that are NOT inherently obvious, and are NOT explained (at least clearly) in any Apple Support or documentation. I would think that the "Synch" button would do that because that's what the word synch (synchronize) means. As far as I can figure out, the "Synch" button (at the bottom right of the main iTunes screen and seen when I select my iPad from the left column in iTunes) only transfers the music files. I have to right-click on the iPad icon to get an option to "transfer purchases from [my] iPad". If I don't do that first, when I upgrade the OS, I lose everything on my iPad that I didn't manually move to the MacBook. Or so the warning box says. Also, after "transferring purchases to the MacBook", how do I know the apps were transferred? When I go to the Apps view in iTunes, the apps are greyed out. So if I click on "Synch", the dialog box asks me if I'm sure I want to do that, because all existing apps and their data on the iPad will be replaced with apps from this iTunes library." So I still don't know how to be confident that all the apps purchased on my iPad that ARE ON the MacBook. So then I have to right-click and select "back-up" and hope there's not some other step that I didn't know about that will result in lost apps or iBook files or who knows what else!