I know that a 'Vorgesetzter' is someone who can give me tasks that I need to do on my job but what's about 'Chef'?
Is my 'Vorgesetzter' also my 'Chef'?
Have both words the same meaning or is only the person who owns the company the 'Chef'?
There is not much difference in meaning when talking about the relation between persons. Yes, "mein Vorgesetzter" is also "mein Chef".
The owner or highest manager is often called something like "Chef der Firma XY" or "Chef von XY". Vorgesetzter is not used like that.
Relations between people:
Herr Müller ist mein Vorgesetzer. / Herr Müller ist mein Chef.
Frau Mayer ist die Vorgesetzte/Chefin von Herrn Müller.
Chef is used in an absolute position more often than "Vorgesetzter":
GDL-Chef Claus Weselsky äußert sich zum Bahnstreik.
In der Rathaus-Schenke steht der Chef selbst hinter dem Tresen.
Kanzleramts-Chef Helge Braun erteilt Spekulation eine Absage.
"Vorgesetzer" is much more formal though.
"Chef" can almost be called informal or simplifying. It's very often used in informal speech or in journalism to talk about hierarchies or positions without using the formal positions of people.
"GDL-Chef" = Vorsitzender der Gewerkschaft GDL
"Chef der Rathaus-Schenke" = Besitzer/Pächter der Rathaus-Schenke (nicht: Chefkoch!)
"Kanzleramts-Chef" = Bundesminister für besondere Aufgaben
Mein Chef kommt aus Indien.
Chef, ich brauch' mehr Geld!
Wikipedia defines "Vorgesetzter" as:
Vorgesetzte sind natürliche Personen, die innerhalb einer Organisation (Unternehmen, öffentliche Verwaltung, Behörde, Militär) mit der Befugnis betraut wurden, Weisungen an nachgeordnetes Personal zu erteilen.
In English:
Supervisors (superiours, line managers) are natural persons who have been entrusted within an organization (company, public administration, authority, military) with the authority to issue instructions to subordinate personnel.
In most companies there is a hierarchy, and not only your boss can tell you what to do, but also your boss's boss their boss and so on, and they also often do it. For example, when the owner of a company decides, that every employee has to wear red jackets and green hats, then this an instruction to which you must obey, even if the owner is not your direct boss.
Also everybody in the line of hierarchy between you and the owner can give you instructions. For example if the department director wants that everybody in the department used toe departments logo on documents that will leave the department, then this is an instruction that neither comes form your direct boss not from the owner, but from a line manager somewhere in the line between them.
All of these people can give you instructions, and in German you call all of them »Vorgesetzter« and »Chef«. In most cases its your direct boss who gives you instructions, so in most cases people talk about their direct boss when they say »mein Chef«. If the context doesn't make it clear, you can also say »mein direkter/unmittelbarar Chef/Vorgesetzter«.