There are a few options, and it may be that you ultimately want to own more than one case, depending on how you use, store, and travel with your guitar.
Here is a roundup of the most common types of cases, and their features.
Lightweight Gig Bag:
This case, while the least expensive of options, offers the least protection for your guitar. It is usually made of nylon fabric and closes with a zipper, and it may or may not have a light layer of padding to offer extra protection to the guitar.
A lightweight gig bag is lightweight, inexpensive, and is better than nothing (using no case at all is not recommended). These cases frequently feature backpack-style straps, making it convenient to carry your guitar around from place to place. It may have an external pocket for keeping music, picks, capos, tuners, etc.
On the downside, it offers very little protection against bumps and bruises, and no structural support for the guitar.
Heavy Duty Gig Bag:
Also made of fabric (heavier nylon or canvas), a heavy duty gig bag will surround the guitar with a thick (1 inch, give or take) layer of foam padding. This case closes with a more rugged zipper, and may also have internal or external compartments for keeping music, picks, capos, tuners, etc.
Heavy duty gig bags tend to be favored by players who may take their instrument from place to place and keep it under close supervision. They are not recommended for planes, trains, busses, band vans full of gear, or any other potentially rough ride. The heavy duty gig bag provides effective protection from minor bumps and bruises, as well as a small amount of structural support for the guitar. Backpack straps make carrying the guitar light and easy.
This is the classic “cheap lightweight” case of years past, before cheaper lighter gig bags became the industry standard. Cardboard cases offer a bit of structural support for the guitar, but usually do not include padding of any kind. They frequently fasten shut by way of three metal clasps, which sometimes have a tendency to pop open and put your guitar at risk of falling out of the case as you are carrying it.
These cases may be a good low-cost option for storing a guitar in a stable environment (in the closet, under the bed), or for minimal travel, such as to a guitar lesson or a friend’s house.
This is the classic sturdy guitar case. Made of wood or fiberglass, a hardshell case also usually features a nicely padded interior, and a snug secure fit for your guitar. The hardshell case provides the best protection for your guitar in a variety of situations, from travel, to safe storage. (not recommended for checked airline baggage, or other “freight” situations however). The tough exterior and snug fit provide excellent protection against bumps and bruises, and good structural support.
On the downside, hardshell cases are heavy and require effort to manipulate and carry with you. They usually include a small interior compartment for picks, capos, tuners, etc., but no larger pocket for music. Backstrap rigs are available to fit onto hardshell cases, to take the weight off your arms when carrying the guitar.
If you absolutely must check a guitar into checked baggage or another freight situation, you should use a flight case. Flight cases are heavy and expensive, but they are designed to provide your precious guitar with the best protection possible.