Multiple trademarks can exist because trademark law originated to protect the use of a given trademark with respect to specific goods or services. The trademark user created those goods or supplied those services and distributed those goods or services in association with a trademark (for goods) or a service mark (for services) which the public thereafter came to understand identified the source (or creator) of the particular goods or services.
Without an identification to particular goods or services, trademarks and service marks do not exist. It is possible, and quite common, for a given trademark to be associated with goods created by one person, and for the exact same trademark to be associated with entirely different goods created by some other person. The key to whether or not those identical marks are in conflict is whether or not the public in general is likely to be confused as to the source or origin of the goods associated with the identical marks. It is unlikely, for example, that the public is going to confuse the use of the word "windows" with respect to a computer operating system with Gienow Windows & Doors with respect to conventional windows in buildings. At the other extreme, a vendor who sought to sell conventional windows in buildings under the name Geenow Windows & Doors might well expect to receive a cease and desist notice from the people at Gienow Windows & Doors.
A virtually limitless number of factual situations can be postulated between these two extremes, and whether or not one mark is in conflict with another is to be determined by one of the most subjective tests in the law: whether or not the public is likely to be confused as to the source or origin of the goods or services in question. To come back to the example, the public will not believe that the Microsoft operating system comes from Gienow Windows & Doors, or that conventional windows in buildings are products of Microsoft (another topic, not covered here, is sheer size - it is more likely that the public would believe that a behemoth like Microsoft had expanded into the windows-for-buildings market than that Gienow was now marketing a computer operation system, but we leave that issue for another time).
Where the question of whether or not this confusion between one product or service and another is answered in the affirmative, trademark law traditionally protected the first party to use the trademark, and this concept survives to this day. Trademark rights are to be determined, generally, in favor of the first user.
This brings us to another wrinkle in trademark law: since the test is the likelihood of confusion, the locality of the market has to be taken into consideration. Assuming Gienow Windows & Doors are not distributed in Japan, it is entirely possible that a Japanese manufacturer could operate under the name Geenow Windows & Doors (or even Gienow Windows & Doors) without confusing the Japanese public who, in the example, simply have no awareness of the existence of the American trademark Gienow Windows & Doors. The same rule applies to Microsoft, except that its "locality" is world-wide.
To better understand the foregoing, let's take a look at the anatomy of a specific (although fictional) trademark infringement case. There will be portions of the story which may be confusing because of the very nature of trademark law, and the reader may not wish to follow every nuance. There will be no test. Hopefully, however, following this story will give the reader a better understanding of a difficult subject. If you happen upon it before it is finished, feel free to bookmark it and come back to additional material later, since new sections will be uploaded as they are completed until the story is fully told. More.