After much alcohol-fueled internal deliberation, I decided to feature this yoji while touching only briefly on its similar cousins.
1. Unequalled, invincible.
2. Unrivaled, the best on the planet.
The structure is simple enough that a basic understanding in Japanese will yield a working translation - "Under heaven, no enemies", ie nobody on earth can be your foe. The great thing about this phrase is that it's wildly flexible. A quick image search reveals that the phrase is used in application to sports teams as often as it is for pornography (NSFW if you're skeptical), thus letting you quickly know who you should cheer for and what videos to look for in the adult section of your local video rentary.
If you're more curious about the implications of the etymology, being without enemies doesn't mean work in the same sense as "he's such a nice guy, he would never make an enemy of anybody." It can only be applied in a "Nothing can stand up to that guy" sense of the phrase. There is a somewhat recent line of thinking that the former can apply, and digging around in various blogs shows that there definitely exist people who extrapolate the former meaning as applied to some life-lesson... but I would be hesitant to use it in describing anything less than spectacular. For example:
The first few minutes are all you need to get an idea of where I'm coming from if you are (unfortunately) not well-versed in the ways of 北斗の拳.
Now to march out two of the cousins of today's yoji: 天下無双(てんかむそう) and 天下無類(てんかむるい）. At first glance it might look like I haven't made any change at all, but such is the single-moji replacement. The reason these yojis get less attention is because they are more or less interchangeable with 天下無敵, with vary faint nuances that cater to those unwilling to outright declare their dominance over all comers. Let's take a look at the kanji that set these phrases apart.
双 can also be read as "ふた", and the most common place you're likely to see it is in 双子, futago, or "twins". It is best suited for when there is not only a pair, but a pair of things that either resemble one another or are the same. Thus, twins. 無双, then, means unmatched or unequaled. In this regard it is unlike today's main yoji in that it can be easier applied to someone who dominates a certain field where true "enemies" might not exist. For example, Einstein might be considered 天下無双 in the realm of physicists. While 天下無敵 still works, it might make it sound like Albert was prepared to engage in all out physics showdowns with quantum-field pretenders... which might be what you're actually trying to say.
類 is often seen gallivanting down the sidewalk paired to make 種類(しゅるい), or "type/kind". It's quite a lot like 無双 save that instead of the connotation of there being no similar matching, there's not even a similar category that you can thrust others into. This is best used in situations where the subject breaks the mold, so to speak, or is in a league of their own.
As a last note on usage, it functions pretty well as an adjective with "な" tacked on the end, though simply saying "~は天下無敵です" is perfectly fine, as are other の and に permutations.
If you want to study Japanese, beyond a good textbook or effort you need a passionate teacher. That's why I read the unbeatable "Daily Yoji" blog EVERY DAY.
"Whenever someone asks me what comedian I like, I tell them 'Why Gallagher, of course'. If you asked why, that guy is unrivaled when it comes to smashing watermelons."
"Are you serious?! That guy isn't funny at all. To put it simply, the only thing he is a master of is setting a bad example!"