Until 2012, Apple sold a $100/yr package called MobileMe. It gave you a mail address at "me.com", the ability to publish web pages and a photo gallery online, file storage, a "Back to my Mac" service, and the ability to share contacts, bookmarks, and calendars with iPhones for about $9/mo.
MobileMe has now been superseded by iCloud. Users of Macs using OS X 10.7.2 or later, or iPhones, or iPads, get a free account. Apple says it "stores your music, photos, documents, and more and wirelessly pushes them to all your devices." Up to 5GB of storage is free; you can buy more capacity: $100/yr gets you 50GB of iCloud storage.
iCloud lets each user set up a mail address at "icloud.com". (If you have a "me.com" address, you can keep it.) "Back to My Mac" provides remote web access to Macs running Lion and Mountain Lion, supporting remote access and screen sharing. Another feature of iCloud is "Find my iPhone," that can locate your iPhone, iPad, or Mac, useful if it gets lost or stolen.
iCloud is integrated with iTunes and allows you to keep your music and purchased books in the cloud, and use them on any of your Apple devices. iCloud is also integrated with Photos, and lets you store your "Photo Streams" in the cloud. (See below.) These services don't count against your storage quota. Documents written by Apple's iWork applications can also be synced to iCloud (sort of like Dropbox, but only for Apple). There is an iOS 5 app, "Find my Friends," that lets family and friends using iPhones, iPads, and iPods find each other, up to 50 friends.
There are two related iCloud concepts for photos: My Photo Stream stores photos you upload to it from your iPhone or iPad for 30 days. Shared Photo Streams are different: you can populate them by downloading My Photo Stream pictures into Photos, and then moving the photos into the shared streams, or you can add other photos to the shared streams. Although these photos do not count against your iCloud storage quota, Apple has set various limits on iCloud photo stream usage: the main one is that each Shared Photo Stream can contain 5000 photos or videos, and you can have 100 streams.
MobileMe's ability to publish web pages, its online photo gallery, and iDisk storage are gone: you can find similar free services from other companies. iCloud does not support syncing of Mac Dock items and System Preferences from your Mac.
You still need a home dialup or broadband ISP account, and it may include some services like storage and mail: for example, Comcast provides both. iCloud does not eliminate the need to back up your computer's hard drive: you should still use Time Machine.