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According to several newspapers, e.g. this article, or even the (now former) Attorney General's resignation letter, President Trump asked Jeff Sessions to resign. Similar things are also shown in TV shows from time to time.

If I remember correctly, the President can fire any cabinet member. So why would he request Sessions to resign instead of just firing him?

In addition to the niceties listed by another answer, this allows Trump to appoint a temporary replacement according to the Federal Vacancies Reform Act of 1998:

The Vacancies Act’s requirements are triggered if an officer serving in an advice and
consent position in the executive branch “dies, resigns, or is otherwise unable to perform the
functions and duties of the office.”

The Vacancies Act: A Legal Overview

It's not clear that this would apply to a situation where he is fired. In the case where it doesn't, the deputy AG, Rod Rosenstein, would assume the acting role, which is probably not what Trump had in mind.

So it's pretty clear by getting his resignation he explicitly maintains the authority to pick his replacement. If he hadn't resigned, this almost certainly would have been challenged, as one government oversight writer says:

[T]he moral hazard created by allowing the president wide discretion to make an unreviewable temporary appointment to act in place of a Senate-confirmed official he fired is one good reason why this omission might have been intentional on Congress’s part. On the other hand, for most positions there is no mechanism to fill a vacancy temporarily other than the VRA, and it would be odd if there were no mechanism whatsoever to fill vacancies that result from a termination pending confirmation of a replacement.

If the Attorney General Is Fired, Who Acts as Attorney General?

It's pretty customary to let someone resign rather than fire in government (or any senior leadership role, really). Even if we suppose that the reason he was asked to resign was because of a very real improper action on Sessions part (i.e. Sessions acted in an inappropriate way for the role and Trump had some legitimate reason to fire him for cause), Sessions is still a valuable employee for a good many private sector cooperation and may even return to run for his own senate seat on the next cycle it's available. Being fired can hurt these prospects when he applies or may leave his next employer with no legitimate way to spin the candidate as a capable employee, because his last boss fired him. Resignation can be spun... yes, most people will read the phrase "was asked to resign today" is read as "he was fired", but on any job application, this is asked in a form of "What was the reason for leaving your last job." If the job seeker writes "I resigned" it implies that the problem was not his fault but his bosses numerous and often well know questionable management style quirks.

Additionally, certain benefits come with resignation that do not come from firing someone. For example, in some employment situations, certain benefits are given in your severance package depending on the question of who initiated the employee leaving (see the Office Episode where one of the guys from the Merger was about to quit, only for Michael to do the "You can't quit, cause you're fired" line... and then realized he screwed the pooch.).

Finally, remember what happened when Trump fired Comey, who had managed to piss of just about everyone in Washington in the past year or so. Comey immediately started to go rogue and drop claims against Trump that he was not doing when he was gainfully employed. If we revisit the possibility that sessions may have actually done something wrong, Trump could offer him to resign to gain a possible... um... insurance (blackmail being such an ugly word and all that...) that Sessions doesn't start talking about his former boss on all the news cameras he can get pointed at him. Trump won't talk about the reasons for his request, and Sessions won't talk about his lousy boss.

And this isn't the only way to do this. Almost any time a major seperation happens in creative industries (the boy band breaks up, the director leaves the film project, or an actor walks off set) expect one of the two participants to cite vague "Creative Difference" as the cause of the seperation... it's best to read as they had a big fight over something (It could even be creative) and one of them was fired (though in music acts, it tends to be the band are too mad to perform... studios fire the directors or actors because of legit purposes but don't want the likely film to be called into doubt as being good by the movie watching public.).