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I believe the <a href="http://www.apple.com/timecapsule/" rel="ugc">Apple Time Capsule</a> is the solution you are looking for. It's not cheap though...
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The lacie stuff is Mac friendly. Get one with FireWire 800 if your Mac supports that. Then allow time machine to use the drive. 
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Any drive is mac compatible. Just buy a decent priced good sized USB drive and use disk utility ( applications, utilities, disk utility) to format it as mac is extended (hfs). Then use time machine ( also in the apps folder) to backup to the drive.
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I bought a 2TB GoFlexDesk external drive, made by Seagate from Costco. I thought I was set for life!<br />
It carried the Mac icon with other icons to indicate compatibility. But it really wasn't easy to connect. I had to get my son to do it with instructions from you!<br />
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I have had some problems, which my local computer store helped me with. She said I should get a hard drive specifically made for Mac. <br />
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Can you advise me specifically what to buy, please, as they all products seem to indicate, with the icon, that the product is Mac-compatible. I want it to automatically back-up. I dont want to think about it!
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I bought a 2TB GoFlexDesk external drive, made by Seagate from Costco. I thought I was set for life!
It carried the Mac icon with other icons to indicate compatibility. But it really wasn't easy to connect. I had to get my son to do it with instructions from you!

I have had some problems, which my local computer store helped me with. She said I should get a hard drive specifically made for Mac. 

Can you advise me specifically what to buy, please, as they all products seem to indicate, with the icon, that the product is Mac-compatible. I want it to automatically back-up. I dont want to think about it!

Advice for buying external hard drive for Mac Book Pro

All of Apple's hardware is produced solely by Apple Inc., who determines internal systems, designs, and prices. Apple buys certain components wholesale from third-party manufacturers. The current Mac product family uses Intel x86-64 processors. All Mac models ship with at least 1 GB RAM as standard. Current Mac computers use ATI Radeon or nVidia GeForce graphics cards and include a dual-function DVD and CD burner, called the SuperDrive. Current Apple hardware includes the iPad, iPhone, iMac, Mac Pro, Macbook, iPod and Apple TV. Discontinued hardware includes the Apple I, Apple II, and the Apple III, the Lisa, the Newton, the Quadra, Performa, Centris, Macintosh II and LC series, the Power Macintosh, iBooks and the MacBook.

Apple Hardware

Storage devices include any device used for storing and retrieving digital information. Hard disk drives (HDDs) use one or more rigid ("hard") rapidly rotating disks (platters) coated with magnetic material. Data is accessed in a random-access manner, meaning that individual blocks of data can be stored or retrieved in any order and not only sequentially. The primary competing technology for secondary storage is flash memory in the form of solid-state drives (SSDs), but HDDs remain the dominant medium for secondary storage due to advantages in price per bit and per-device recording capacity. CD-RW (Compact Disc-ReWritable) is a digital optical disc storage format that allows information to be stored outside the mechanical HDD.