Obviously, it's the middle one. You can know this because the lamp is on (you can see its light), and because you can see where the other two cords begin. You know this even though you can see neither the outlet nor the lamp. Sheer process of elimination is enough here.
Understanding the Church works somewhat similarly. The true Church must go back to the Person of Christ Himself, He's the power outlet, so to speak, of grace. He promised to establish His own Church, rather than simply leaving the task up to His followers, and He promised that the gates of Hell would not overcome that Church (Matthew 16:18). We'll ignore for now that Christ promised to establish this Church upon Peter, the first pope. The simple matter that we should be looking for Christ's own Church is sufficient.
The early Church Fathers understood this quite well. For example, in Against Heresies, written in 180 A.D., St. Irenaeus wrote:
For now, let's also ignore the fact that Irenaeus just brought up the centrality of the Bishop of Rome, the pope, and the fact that he called it “a matter of necessity that every Church should agree with this Church, on account of its preeminent authority.” Here's a more fundamental question: why is Irenaeus producing this list in the first place?2. Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who, in whatever manner, whether by an evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say,] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. For it is a matter of necessity that every Church should agree with this Church, on account of its preeminent authority, that is, the faithful everywhere, inasmuch as the tradition has been preserved continuously by those [faithful men] who exist everywhere.
3. The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric. This man, as he had seen the blessed apostles, and had been conversant with them, might be said to have the preaching of the apostles still echoing [in his ears], and their traditions before his eyes. Nor was he alone [in this], for there were many still remaining who had received instructions from the apostles. In the time of this Clement, no small dissension having occurred among the brethren at Corinth, the Church in Rome dispatched a most powerful letter to the Corinthians, exhorting them to peace, renewing their faith, and declaring the tradition which it had lately received from the apostles, proclaiming the one God, omnipotent, the Maker of heaven and earth, the Creator of man, who brought on the deluge, and called Abraham, who led the people from the land of Egypt, spoke with Moses, set forth the law, sent the prophets, and who has prepared fire for the devil and his angels. From this document, whosoever chooses to do so, may learn that He, the Father of our Lord Jesus Christ, was preached by the Churches, and may also understand the tradition of the Church, since this Epistle is of older date than these men who are now propagating falsehood, and who conjure into existence another god beyond the Creator and the Maker of all existing things. To this Clement there succeeded Evaristus. Alexander followed Evaristus; then, sixth from the apostles, Sixtus was appointed; after him, Telephorus, who was gloriously martyred; then Hyginus; after him, Pius; then after him, Anicetus. Soter having succeeded Anicetus, Eleutherius does now, in the twelfth place from the apostles, hold the inheritance of the episcopate. In this order, and by this succession, the ecclesiastical tradition from the apostles, and the preaching of the truth, have come down to us. And this is most abundant proof that there is one and the same vivifying faith, which has been preserved in the Church from the apostles until now, and handed down in truth.
The answer is that he's showing that the true Church can trace its lineage all the way back to St. Peter, and thus, back to Christ Personally. This is why the Nicene Creed declares our belief in “One, Holy, and Apostolic Church.”
Any group of Christians can claim to be the true descendants of Christ, in the sense that they (rather than anyone else) really get what He was talking about, and that they're carrying on His message. We've heard competing visions of Christ from a God of Wrath to Jesus Christ, Superstar.
The only way to know who's right is by following that cord, Apostolic succession, back to the Apostles, and thus, back to Christ. This is the system the Apostles set up. St. Paul says to his student, St. Timothy, “And the things you have heard me say in the presence of many witnesses entrust to reliable people who will also be qualified to teach others” (2 Timothy 2:2). So we can see the line forming already: from Paul to Timothy, from Timothy to those reliable people who Timothy entrusts the Gospel, and so on.
This is something different from simply proclaiming the Gospel, which Paul and Timothy did to everyone, reliable or not. This is a matter of passing on the faith through discreet individuals, who will then take up the mantle as teachers, rather than simply believers. Remember that it was also St. Paul who asked, “And how can anyone preach unless they are sent?” (Romans 10:15).
You can't just become a preacher because you love the Gospel -- you also have to be sent by the Church. Plenty of people love the Gospel, but misunderstand core teachings -- requiring the Church's permission is an important step in “quality control.”
When this practice wasn't followed, the Apostles denounced the men who went out on their own (Acts 15:24), and sent Paul and Barnabas in their place (Acts 15:25). So this notion of each generation of Church leadership selecting the next generation and commissioning them is a solidly Biblical one.
Apostolic Succession, then, in the power cord. It should show a clear and visible lineage from Jesus Christ to the person standing before you preaching. And to point out the obvious, if it doesn't go all the way back to the power source, it's worthless. In the picture above, the top cord goes back much further than the bottom one, but since neither connects with the power source, it doesn't matter.
Likewise, Reformed Christians often look down upon our Evangelical brethren, because Evangelical traditions date back to the 19th and 20th century, while Reformed traditions date back to the 16th and 17th century. But since neither of them can trace Apostolic Succession back to the first century, back to the Apostles, back to Christ during His earthly ministry, who cares?
St. Jerome, who many Protestants feign allegiance to on the question of the canon of Scripture, was quite clear on how to distinguish the true Church from false ones:
We ought to remain in that Church which was founded by the Apostles and continues to this day. If ever you hear of any that are called Christians taking their name not from the Lord Jesus Christ, but from some other, for instance, Marcionites, Valentinians, Men of the mountain or the plain, you may be sure that you have there not the Church of Christ, but the synagogue of Antichrist. For the fact that they took their rise after the foundation of the Church is proof that they are those whose coming the Apostle foretold. And let them not flatter themselves if they think they have Scripture authority for their assertions, since the devil himself quoted Scripture, and the essence of the Scriptures is not the letter, but the meaning. Otherwise, if we follow the letter, we too can concoct a new dogma and assert that such persons as wear shoes and have two coats must not be received into the Church.We don't have to wonder, then, what Jerome would have thought of men calling themselves Lutherans or Calvinists: he already told us. Nor is there any question what he'd say to those claiming to follow sola Scriptura, and freely concocting new dogmas that they derive from their own reading of Scripture. But most importantly, look at how clearly Jerome condemns those who start new denominations.
If anything, Jerome is too rough on the heretics of his day, by denouncing their churches as synagogues of the Antichrist (a reference to Rev. 2:9 and Rev. 3:9). But there should be no question that the early Christians found the legitimate Church by tracing Apostolic succession. And there's no question that they rejected any churches founded after the time of the Apostles, even if that church claimed to have new insights from Scripture.
All of this stacks up to a compelling case for the Catholic Church. We can say quite definitively when each Protestant denomination started, including Anglicanism (which began in 1534, when King Henry VIII declared himself the head of the Church of England). But with the Catholic Church, seemingly no two Protestant apologists can agree on who the first pope was, if not St. Peter.
This brings me to my last point: that Apostolic Succession is not the only criterion we're looking for. The Orthodox can also plausibly claim to have Apostolic Succession, as can the Anglicans (albeit less plausibly). As Catholics, we acknowledge the Orthodox as part of the Church for this very reason. But how do we know, then, whether Catholicism or Orthodoxy is the correct route to follow?
Well, look to the parts I glossed over above. What else does Jesus tell us in Matthew 16:18? That He'll build His Church upon Peter. And what else does Irenaeus tell us? That Peter, along with Paul, founded the Church of Rome.
And suddenly, a lot of things start to jump out. For example, the fact that Irenaeus (the bishop of Lyons) traces the pope's lineage, rather than his own -- that suggests that the pope has a pretty special place in the Church. And then we notice a whole lot of other people doing the same thing: e.g., St. Optatus of Milevis gives a similar list elsewhere, as does the Liberian catalogue of 354 A.D., as does the early Church historian Eusebius.
Each one traces the lineage of Peter in Rome, even though Irenaeus is clear that they could trace anyone's. So we should be looking for two related features: Apostolic Succession and communion with the Bishop of Rome. These two things are tied together, and found only in one Church on Earth, the Catholic Church.