I am writing to find out the best option to amplify the existing WiFi signal in my house in order to be able to be online in my backyard with my MacBook Air. Would purchasing an AirPort Express be a good solution, or a simple extender/repeater would do the job?
An Access Point would be sufficient - & cheapest.
[just some examples from Amazon, no recommendations specifically. Note, the filter on Amazon isn't great, there are other device types in there, as well as simple Access Points]
An Access Point needs no additional functionality other than being able to push a signal further.
You wire it via your existing Ethernet to where the signal will reach the garden. All routing/DHCP functionality is then still provided by your existing router.
If you can't, for any reason, get an Ethernet cable far enough to reach, then you would need a Range Extender, which is in essence the same thing, but wireless throughout. I'd still go for an AP as first choice, if at all possible.