The traditional way of carrying things on a bike. The basket, whilst not everyone's cup of tea, is an excellent way of carry your daily gear on a bike. You can just drop your normal bag inside the basket and go, with the added advantage of being able to keep your stuff in sight at all times.
However, if it rains, whatever is in the basket will get wet, so make sure everything is appropriately bagged up against the elements. A top tip is to buy an elastic luggage net for the top of your basket, to stop stuff bouncing out of the basket over the bumps.
Most people own a backpack of some sort and they're ideal for transport commuter sized loads - i.e. a laptop, lunch, tools and waterproofs. Outdoor pursuits type packs are great, as they're usually water resistant and often have special back systems to help prevent sweaty back syndrome, plus some have ‘sternum' straps to keep the bag nice and stable on your back.
Proper courier bags sling across one shoulder and are stabilised by a second strap around the waist. Courier bags have the advantage of carrying weight lower down (lower centre of gravity is always good on a bike) and also resting most of the weight on the lower back/hip area, avoiding the aforementioned sweaty back syndrome. Beware fake courier designs with no stabiliser strap - a fully loaded courier bag swinging loose mid-traffic is a scary thing indeed.
These are bags which mount on each side of a pannier rack, which mounts on the rear of the bike. If you prefer to let the bike carry the weight (rather than you), panniers are ideal. They carry weight low to the ground, offer great capacity for all your daily goods, and the best designs attach securely and detach quickly thanks to smartly designed quick release attachments.
Other models will convert from rucksack to pannier for ultimate versatility. However, many bikes cannot accept a pannier rack - some bikes don't have the necessary attachment points and disc brakes can cause fitting problems on others. If in doubt, check with your local bike shop before taking the plunge.
The choice of the traditionalist - these bags mount laterally behind the saddle, requiring saddle loops (as found on Brooks saddles and others) or requiring a specific support. They're great for commuting sized loads, don't require a rear rack (like panniers) and they're also up out of the way of road grime and potential damage.
Do not dangle carrier bags off your handlebars! Whatever method you choose you'll avoid the perilous pursuit of dangling shopping bags on your handlebars. This seriously destabilises your bike and makes it difficult to turn or react quickly to hazards. Choose one of the methods above and you'll be fine.