From Wikipedia

A modern melodrama is a dramatic work in which the plot, typically sensationalized and for a strong emotional appeal, takes precedence over detailed characterization. Melodramas typically concentrate on dialogue that is often bombastic or excessively sentimental, rather than action. Characters are often flat, and written to fulfill stereotypes.

Contemporarily, we use the word melodrama for narrative works that go to great lengths to induce certain kinds of emotion at all costs, to detriment of the internal cohesion of both plot and characterization, often in a manner that some consider cheesy, corny, or excessive. All soap operas are melodramas, as are many movies and TV shows. Some melodramas are cheap and fail to achieve their effects, while others can be more rich and even sophisticated. In a way, many mainstream stories are, to some extent, melodramas, even when there are other, more salient genres. There's melodrama in action, crime, and science fiction.

Here are some examples of what I consider more or less contemporary melodramas:

• The Young and the Restless
• Grey's Anatomy
• The Color Purple
• Downton Abbey
• Dawson's Creek
• Avatar
• Gone with the Wind

And here are some stories that are not melodrama, but contain a whole lot of it:

• True Blood
• The Walking Dead
• A.I. Artificial Intelligence
• Pretty much every super-hero CW show
• Pretty much anything in the MCU

start rant

First, my personal definition of the term: melodrama is a narrative that appeals to our stronger emotions in a lengthy, recurrent, unjustified and exaggerated fashion. Unlike drama, which plays to your sentiments in a more contained and psychologically realistic manner, melodrama overwhelms us with every trick in the book to elicit a powerful emotional reaction by any means necessary.

You can tell from my phrasing that I'm not a fan of the genre, but that's beside the point. Melodrama has its place: operas and soap-operas wouldn't exist without it, and, in moderation, it's a practical way to inject emotion in plots that would be otherwise hermetic and dry.

But even sweetness in excess will make you vomit, and many interesting productions exaggerate it to the point of nausea. Arrival is awesome, but did Amy Adams character (which was basically one the smartest persons on Earth) really need to spend so much time as a freaking wife? We had the coolest movie aliens in the last 20 years, did she really need to marry a boring physicist? And what about the whole parenting conundrum in Interstellar? You're in fucking space, I couldn't care less about your failings as a father! No one could save 1998s Armageddon, but the struggle to explode the giant asteroid heading towards the Earth was way more interesting than Liv Tyler saying goodbye to Bruce Willis over some corny Aerosmith song. The TV show The Killing was particularly annoying... what would prefer, awesome investigation scenes with constant new developments or 30 versions of "look how the same family is grieving in a slightly different way"?

But credit where credit is due: some moviemakers know a thing or two about concision. So my props to Fernando Meirelles (City of God), José Padilha (Elite Squad), Alfred Hitchcock, David Fincher, Sidney Lumet, Martin Scorsese, Chad Stahelski (from John Wick!!!!) and many others. Thank you for not wasting my time!

EDIT1: And just make things perfectly clear: my issue is not with the presence of drama or melodrama, but with its amount...
EDIT2: to be even more clear: this does not mean that I wish for all movies to be sterile, dry or devoid of emotional content...
EDIT3: a lot of answers seem to ignore the differences between drama and melodrama, the previous edits and the nuance of the post. Ahh... what can I do? :P

end rant