There are an estimated 10.000 film festivals around the globe to which you can submit films to. Ranging from major A list festivals like Cannes and Sundance, to obscure little festivals in the back of a local bar. Often it is hard to see the difference and some research is generally required. Most festivals have switched to allowing you to submit via on-line submission platforms. Below is a list of online platforms:
https://www.withoutabox.com, https://www.clickforfestivals.com, https://festival.movibeta.com, https://www.reelport.com, https://www.shortfilmdepot.com, https://festhome.com, https://filmfreeway.com, https://www.filmfestplatform.com, Up to Fest, Festivals.es, http://www.filmfestivalsalliance.org/list_of_festivals.htm
Withoutabox (WAB) is by far the largest platform and was purchased by IMDB some years ago, which was itself purchased by Amazon. WAB also tends to charge significantly more fees for submission (partly driven by the festivals), with fees that range from 30-90 USD. Submitting a film through WAB is the easiest way to get a film listed on IMDB, which is one of the premiere ways to build an online status for film makers.
Other platforms tend to list more festivals with no fees or a nominal fee (2-3 EUR) for the platform and therefore may be a good starting point to test your film's reception before investing large amounts of money into submitting to many expensive festivals.
Apart from a few A list festivals (Cannes, Berlin, Venice, Locarno), few festivals require premiere status to submit (although it may weigh in the selection). Most large US festivals (Sundance, Tribeca) will only require premiere status in their local urban area or state.
Some of the A list festivals maintain their own online submission forms, like for example the Cannes Film Fest: http://www.cannescourtmetrage.com. Cannes also hosts the Short Film Corner and every year accepts around 1200 films. There is no public screening in the corner, but acceptance does grant you cheap access to the otherwise difficult to access festival, facilities to organize your own mini-screening, attend red-carper events, attend workshops, network, and generally learn the film business in what is arguably the largest annual film market.
Competition is fierce and some times it may be a competitive edge to submit to a festival that has not switched to on-line submission yet. They are hard to find, but also more likely these festivals charge no submission fees. Some resources for festivals are:
A list of barcelona based festivals: http://www.barcelonafilmfestivals.com/
Wasabi Creation is a group that helps facilitate film submissions to Japanese Film Festivals.
Apart from Film Festivals, there are also Film Contests. Have a look here to find a selection: https://www.onlinevideocontests.com/
Submission formats range per festival. Few i.e. pretty much none, still ask for a film copy. Most smaller festivals ask for a video file, or DVD/Blue Ray.
Larger Festivals will require a DCP, which is the digital cinema format that has replaced the 35mm prints of analog projections. Making a DCP is a complex affair and was until recently an expensive last step in creation screening copies of your films. However, open source projects have been under way to make DCPs more available for filmmakers working on small budgets: