The "spirit of this world" refers to the unrepentant sinfulness of people, especially those inclinations that lead people to rally against and undermine the Church. The "acting body" of the spirit of this world seems to be primarily human (human impiety, malice, and so on), though it is certainly stoked and sustained by the forces of darkness and the devil in particular. This "spirit of the world" does not specifically refer to the devil, who would be first among the forces of darkness.
A bit of history is now in order (at least for those interested in the details).
Blessed John Paul II is speaking about the prayer to St Michael the Archangel. This prayer was composed by Pope Leo XIII, and it is fairly reasonable to assume that Bl John Paul understands the background well enough to know what his predecessor was getting at. The prayer is a plea for aid in defending the Catholic Church from her enemies (a prayer similar to, for example, Lamentations 3). It was added at the end of the Mass in 1886. Here is a relevant excerpt from the longer version of this prayer:
[...] That cruel, ancient serpent, who is called the devil or Satan who seduces the whole world, was cast into the abyss with his angels. Behold, this primeval enemy and slayer of men has taken courage. Transformed into an angel of light, he wanders about with all the multitude of wicked spirits, invading the earth in order to blot out the name of God and of His Christ, to seize upon, slay and cast into eternal perdition souls destined for the crown of eternal glory. This wicked dragon pours out, as a most impure flood, the venom of his malice on men of depraved mind and corrupt heart, the spirit of lying, of impiety, of blasphemy, and the pestilent breath of impurity, and of every vice and iniquity.
Be favorable to Thy Church, the Bride of the Lamb without spot, whose enemies have filled to overflowing with gall and inebriated with wormwood. [...]
Pope Leo XIII issued 85 Encyclicals. If you look at that list, you'll see that a fair number of them are vocal against various secular evils, which took root as Communism, Freemasonry, Nihilism, Socialism, and so on. The first of these encyclicals was Inscrutabili Dei Consilio, On the Evils of Society, which closely mirrors the above prayer:
[...] For, from the very beginning of Our pontificate, the sad sight has presented itself to Us of the evils by which the human race is oppressed on every side: the widespread subversion of the primary truths on which, as on its foundations, human society is based; the obstinacy of mind that will not brook any authority however lawful; the endless sources of disagreement, whence arrive civil strife, and ruthless war and bloodshed; the contempt of law which molds characters and is the shield of righteousness; the insatiable craving for things perishable, with complete forgetfulness of things eternal, leading up to the desperate madness whereby so many wretched beings, in all directions, scruple not to lay violent hands upon themselves; the reckless mismanagement, waste, and misappropriation of the public funds; the shamelessness of those who, full of treachery, make semblance of being champions of country, of freedom, and every kind of right; in fine, the deadly kind of plague which infects in its inmost recesses, allowing it no respite and foreboding ever fresh disturbances and final disaster.
3. Now, the source of these evils lies chiefly, We are convinced, in this, that the holy and venerable authority of the Church, which in God's name rules mankind, upholding and defending all lawful authority, has been despised and set aside. The enemies of public order, being fully aware of this, have thought nothing better suited to destroy the foundations of society than to make an unflagging attack upon the Church of God, to bring her into discredit and odium by spreading infamous calumnies and accusing her of being opposed to genuine progress. [...]
Sound familiar? Well, it does to Catholics, and of course to all who would rightly call themselves Christians, as we will soon see. Note that this was written in 1878. This was before the first World War, before anti-religious Communism as we tend to remember it. Now, later in this same encyclical, Pope Leo XIII directly references Col 2:8, and we find ourselves going further back in history, to Paul:
See to it that no one captivate you with an empty, seductive philosophy according to human tradition, according to the elemental powers of the world and not according to Christ.
And here is 1 Corinthians 2:
6 Yet we do speak a wisdom to those who are mature, but not a wisdom of this age, nor of the rulers of this age who are passing away. [...] 12 We have not received the spirit of the world but the Spirit that is from God, so that we may understand the things freely given us by God.
Of course, "world" has a number of meanings throughout the Bible (creation, fallen humanity as in John 3:17). Here, it refers to the ideas expressed vividly by Jesus in John 15:18--:
The World’s Hatred: “If the world hates you, realize that it hated me first.
If you belonged to the world, the world would love its own; but because you do not belong to the world, and I have chosen you out of the world, the world hates you.
Remember the word I spoke to you, ‘No slave is greater than his master.’ If they persecuted me, they will also persecute you. If they kept my word, they will also keep yours.
And they will do all these things to you on account of my name, because [...]
This is what the prayer by Pope Leo XIII refers to, and what Bl John Paul II is speaking of when he says "the spirit of this world".
When we say the Prayer to St Michael the Archangel (defend us in battle, be our protection against...), many of us may have in mind the more-personal evils that we are worried are arrayed against us (which is good and right, since it serves to keep us on the narrow way, as God intends). When the Pope says the prayer, he has evils more vast in mind; he says it as a plea to God for the protection of the entire Church and world.