There are many different Transformers stories, but there are some common threads that run through many of them. Here are those main, most fundamental elements:
The planet Cybertron is inhabited by robotic lifeforms called Transformers. The Transformers are split into political or ideological factions, and have been at war with each other for a very long time. Millions of years, in most of the stories. The factions are usually called Autobots (good guys) and Decepticons (bad guys), but not always.
Many Transformers stories involve an evil, all-powerful being called Unicron. Unicron is titanically enormous: he has a humanoid robot form like the Transformers do, but his other form is a planet. A whole planet. And he eats other planets. In some stories there is a dichotomy between Unicron and a being called Primus. In stories where he exists, Primus is the creator-god of the Transformer race, and is the eternal enemy of Unicron.
There is also usually something called "The Matrix". Sometimes it is a source of the Transformers' "lifeforce". Sometimes it's a powerful device wielded by the Autobot leader. And sometimes the Matrix is intimately connected to Primus in some way. (Or any combination of these.)
Along the lines of what I said about the stories above, there are some characters who appear in many TF toylines. Or, rather, "character types". For example, almost every series has an Optimus Prime and a Megatron who serve as the good and evil leaders. Although characters with these names keep showing up, they are not always the same characters. They usually have similarities, but even the extent of that similarity varies. For example, the Megatron of Robots In Disguise has almost nothing in common with the Megatron of Beast Wars aside from the name and the position of "leader of the bad guys in the story". Their methods, goals, personality, and position within the overall bad guy hierarchy of their stories all differ.
Another character type who shows up more than once is Starscream -- a high-ranking member of Megatron's army, Starscream is ambitious, treacherous, very skilled, and wants to usurp Megatron's position. That's true for all Starscreams that have been presented to us, but they have differences as well. G1 Starscream was very self-involved whereas Armada Starscream showed great affection for other characters and even sacrificed his own life in hopes of helping to save Cybertron from Unicron. (There was also a treacherous Starscream in BW2.)
Aside from Prime, Megatron, and Starscream, there are just a few others who show up repeatedly in similar ways. One is Ultra Magnus. Magnus is a good guy, sometimes with a connection of some sort to Prime. (They are sometimes portrayed as brothers, or as brother-like. Sometimes just as old friends.) Magnus is a good leader, although not as good as Prime. In some stories he wants to lead, and in others he doesn't. The Magnus of the G1 cartoon preferred to serve as an advisor, and was very conservative and professional. The Magnus of the RID cartoon was deeply envious of Prime's position, wanting to take it by force. He was hot-headed and unorthodox. Magnus in the G1 comics had very little interaction with Optimus Prime, but he was an accomplished leader of forces on Cybertron while Prime was on Earth, as well as an almost-unbeatable soldier. Magnus in the Dreamwave-G1 comics seemed to be a bit of a mixture of the G1 cartoon and comic versions.
Also, it's become something of a pattern for characters named Megatron to be reborn into stronger forms named Galvatron, although this doesn't always occur. Primus and Unicron are, in some ways, characters, but they're more like plot devices and are described (briefly) in the previous question.
Not all characters with previously-used names bear any resemblence to the originals, though. For example, RID Grimlock and G1 Grimlock are completely orthogonal to each other. They're not opposites, they just don't have anything in common (aside from being physically strong). Armada Red Alert and G1 Red Alert, too, have no similarities in personality. There have been a bunch of different characters named Prowl and Mirage, too, who have had very little in common aside from sometimes being a police car and a race car, respectively.
If you want to know about important characters for a particular story, rather than to the mythos as a whole, just check the section of the FAQ devoted to that story.
Transformers toys are designed through a collaboration of Hasbro in the United States and Takara in Japan. The very first Transformers toys were entirely the work of Takara, having been released in Japan a few years prior under other brand names. Starting with some of the 1986 toys (the movie characters in particular), design work was shared between Takara and Hasbro. Toys that Takara released exclusively in Japan were presumably designed entirely by Takara without input from Hasbro. The working relationship between the two companies has grown over the years, to the point that Armada was said to have been entirely co-developed. In contrast, when it came to something like Beast Wars, the toys were primarily designed with Hasbro's desires in mind and Takara mainly providing the engineers. Takara later opted to release BW in Japan as well, but it was designed with Hasbro's sensibilities in the forefront. Compare the Hasbro-style animal Transformer designs in BW with the Takara-style designs in BWN to start to get a feel for how they differ.
Manufacturing of all Transformers toys is handled by Takara. Hasbro does their TF manufacturing through Takara at Takara's factories in China.
There have been a lot of cartoons coming from different sources.
The original cartoon series was produced by Marvel Productions under contract from Sunbow Productions. (Sunbow is a company that was established by Hasbro's advertising agency, Griffin Bacal, for the purpose of producing animation to sell toys.) The writing, voice acting, and most of the design work for the show was handled by Marvel Productions. The animation itself was done by a number of studios in Japan and Korea, although the vast majority of episodes were animated by either Toei or AKOM. The Transformers movie was also produced by Sunbow, Marvel, and Toei. After the American G1 cartoon ended, Takara produced more cartoons with Toei to broadcast in Japan.
Beast Wars and Beast Machines were produced by Mainframe Entertainment, a company which specializes in computer animation. Voice acting was handled by groups in Vancouver -- casting by BLT Productions, and recording at Pinewood Sound.
The cartoon for Robots In Disguise was a dub of the Japanese Car Robots. The animation itself was done by DongWoo Animation in Korea. The English dub was produced by Saban Entertainment (which no longer exists).
Most recently, Armada and Energon were both produced by a Japanese company called We've (previously known as Aeon). Armada was animated for Aeon/We've by a studio called Actas, and Energon was animated by Actas and A-Cat (for the CGI parts). The english dubs of these were handled by Voicebox, another Vancouver-based recording studio. (Hence the presence of several Beast-era VAs in these series, such as Gary Chalk and David Kaye playing these new versions of Optimus Prime and Megatron.) Also, incidentally, We've/Aeon is also the company that produced the Japan-exclusive BW2 and BWN cartoons.
Not all that different, really. There are some terminology differences, some story differences, and some differences in the released toys.
On the terminology front, all the Transformers lines have different names in Japan and the US. They weren't just translated from one language to the other; they're very different. Within each line there are also some differences. The American "Autobots" are usually referred to "Cybertrons", and the "Decepticons" as "Destrons". These names, or variations on them, are used in every Japanese line, even Beast Wars. (Some minor exceptions: For BWN, the transliterations were slightly altered to "Cybertoron" and "Destoron". In CR, it was "Destronger".) The group referred to as "Decepticons" in RID don't seem to have had a definitive groupname in CR, although if it was anything, it would be "Combatrons", a name which applies, at the least, to the commandos/Combaticons. "Mini-Cons" are called "Microns". And, you may be wondering how the Autobots can be called Cybertrons, considering the name of their home planet. Well, the Japanese name for that planet is Seibertron, not Cybertron. (Pronounced "say"-ber-tron.)
The major story difference between the Japanese and American/international versions of Transformers is that there is more Japanese story. There have been several Japan-exclusive cartoons, some attached to toylines that were Japan-exclusive as well. (See question