What do you want to read? New comics? Back issues? Comic strips?
Where do you want to read it? Your computer? Your tablet? Your smart phone?
The first digital comics were produced on the computer using software (instead of traditional pen & ink on paper) but were still printed and distributed traditionally. Wikipedia has a good history of digital comics.
Today when people talk about digital comics they're more likely talking about how they are consumed by the reader instead of how they are produced. Most comics nowadays have at least some of their production done on the computer. Even those that are drawn "old school" on paper are ultimately scanned and typically lettered & colored on the computer. More and more artists are abandoning pencil, pen, and paper and producing their comics completely on a computer. For now, let's concentrate on ways to consume comics digitally.
Lots of comics material is available in digital form. There are traditional comics available in digital format and comics produced to be viewed on the internet. You can read most current comics on the internet, most are even available digitally on the same day that they go on sale in the comic shops. There are also pirated copies of almost every comic published available somewhere on the internet, but we won't go into details on that here. There are also lots of back issues available, for individual purchase as well as unlimited subscription models with a monthly fee. Older comics in the public domain are also available for free.
Many digital content providers (most notably Comixology) provide either streaming access to digital content or downloads directly into their App in a proprietary format, so the comics are not available for the reader to take and use however they want.
Another popular format is the comic book archive/reader (.cba, .cbr, .cbt, .cbz, .cb7), this typically is a single file containing many individual image files that can be read using viewer software. More details on this format can be found here.
ePub (electronic publication) is a free and open e-book standard by the International Digital Publishing Forum, more details can be found here.
The most portable format is Adobe's Portable Document Format, more commonly known as PDF. This is an open standard document format that can be read by most applications, browsers and operating systems.
Many publishers have their own digital comic portals. Where applicable, these are linked on the Publishers page.
When buying comics, consider what format you want. Are you happy to have them hosted in an App or do you want to be able to download a file to your machine and manage it yourself? Do you want to read new comics the day they come out (most cost the same price as buying a physical copy on the day of release) or are you happy to read them later?
The Individual purchase model has readers browse the available library and select issues that interest them for individual purchase. Digital platforms that make comics available for individual purchase often have deals where #1 issues are free or "bundles" of issues are available at a reduced rate. You can also wait for sales where comics can be had for 99 cents.
Comixology is the most popular digital provider. They support PCs, Mac, iPhone/iPad, Android, KindleFire, and Windows 8. Many of the major publishers are partnered with Comixology and the digital sites of Marvel, DC, IDW, Image, and many others are powered by Comixology. They have a lot of free comic samples and if you buy on one platform, you can access your account/comics library from any other platform.
Madefire combines motion, sound effects and music to enhance the comics reading experience. There is a Motion Book Tool that can be used from your browser but the experience is much better on an iPad, iPhone, and other iOS devices.
comicsplus (powered by iVerse Media) is targeted at Apple's iOS and is meant for use on iPad, iPhone, and iTouch devices.
DriveThru Comics specializes in independent comics, providing popular formats such as CBZ, ePub, and PDF for publishers like 2000AD, AAM Markosia, Airship Entertainment, Archaia, Bluewater, Caliber, Dork Storm, KenzerCo, Top Cow, White Wolf, and others.
CerebusDownloads currently features the issues in the High Society storyline (the 25 issue storyline that put Cerebus on the map as a true indie comics sensation) for 99 cents each. For that price you can download the comic in PDF, CBZ, ePub AND in audio/video MP4 format, where Dave Sim himself reads the entire comic to you as the panels & pages scroll by! There is also an MP4 audio commentary for each issue.
The subscription model has the reader pay a monthly (or yearly fee) to read all the comics you want as long as your subscription is active. The actual number of comics is going to depend on the size of the library of the content provider, but this can be a very low cost way to read LOTS of comics.
Comixology, in addition to offering "buy issues to own", now has an unlimited subscription model for $5.99 per month, and offers a 30 day free trial.
Marvel Digital Comics Unlimited is a subscription service that for a monthly (or yearly) fee gives you unlimited access to the 1000s of Marvel comics available in their library, from the earliest days of Marvel to about 6 months after they are released in stores (though not every comic from 6 months ago will be available). You can check out a review of this service by Dave Frank.
Scribd. offers lots of comics from top publishers Marvel, Archie, Dynamite, IDW, Top Cow, Top Shelf, Valiant, and Zenescope for a single monthly fee. In addition to comics, you also have access to prose books from a wide variety of popular authors and audiobooks all for the same monthly fee. They also offer a 1 month free trial.
ComicBlitz offers 3 modes of use: They have free sample comics, a plan with up to 10 issues per month for $3.99, and ulimited free comics for $7.99 per month.
Many sites offer "teaser" comics for free, usually a 1st issue or some other jumping on point to get you hooked on a series in the hopes that you'll buy subsequent issues. You can find these on many of the sites linked above.
In addition to the free sample comics on the various pay sites, there are a couple of sites with totally free digital comics, albeit older public domain comics. You can find newer bootleg/pirated comics on the internet, but ComicSpectrum does not support and will not link to pirate sites.
ComicBook+ has a tremendous number of old Golden and Silver Age comics that are in the public domain scanned in and available for reading on their site.
The Digital Comic Museum is a great site for downloading free public domain Golden Age Comics. All files have been researched by their staff and users to make sure they are copyright free and in the public domain.