What are vows? Are they relevant even today?

According to dictionary in the NKJ Bible, Vine's Expository Reference Edition,

A. Verb nadar "to vow," i.e., to make a promise to another, with sanctions for not completing the promise.
B. Noun neder, "vow; votive offering."

The vow has two basic forms, the unconditional and the conditional.
The unconditional is an "oath" where someone binds himself without expecting anything in return, Psalm 116:14.
The conditional "vow" generally had a preceding clause before the oath giving the conditions which had to come to pass before the "vow" became valid, Genesis 28:20-22.

Numbers 30 deals with the law concerning vows.

In reference to Acts 18:18 (NKJV) which states

So Paul still remained a good while. Then he took leave of the brethren and sailed for Syria, and Priscilla and Aquila were with him. He had his hair cut off at Cenchrea, for he had taken a vow.

The Life Application Study Bible, KJV, gives the following explanation for this verse:

This vow Paul took was probably a temporary Nazarite vow that ended with shaving of the head and offering the hair as a sarifice (Numbers 6:18).

Numbers 6:18 (NKJV) reads;

Then the Nazirite shall shave his consecrated head at the door of the tabernacle of meeting, and shall take the hair from his consecrated head and put it on the fire which is under the sacrifice of the peace offering.

However, the Old Testament allows for "redeeming" the "vow"; by payment of an equal amount in silver, a person, a field, or a house dedicated by "vow" to the Lord could be redeemed, Leviticus 27:1-25.

This practice declined in Jesus' time. The "votive offering," Ezra 7:16, is also a kind of thank offering, Nahum 1:15. Here even Gentiles expressed their thanks to God presumably with a gift promised upon condition of deliverance, cf. Numbers 21:1-3.

If one chooses to take a vow before God to promise to "love and cherish" during a wedding ceremony, for instance, this is not a conditional vow. One isn't asking for something in return, but simply making a promise before and to God. These types of vows do have a place these days.

However, Jesus warns against oaths in Matthew 5:33-36.

Conditional vows and/or oaths, those vows made expecting something in return would seem contrary to a true heart of worship and love for God. My belief is that God does not want conditional praise, conditional worship, conditional love or conditional promises.

Well, I suggest you read A Defense of Rash Vows by G.K. Chesterton for more enlightement.

Modern sages offer to the lover, with an ill-flavoured grin, the largest liberties and the fullest irresponsibility; but they do not respect him as the old Church respected him; they do not write his oath upon the heavens, as the record of his highest moment. They give him every liberty except the liberty to sell his liberty, which is the only one that he wants.

But here's the gist of it.

1. Married couples make a vow to each other, with Christ as their witness

2. Priests and others make vows to the Catholic Church or consecrate themselves to Mary (as I have).

3. Other vows can be of a more mundane purpose (I promise to eat my weight in kelp), but still important to keep if for no other reason than, 'to thine own self be true'

The man who makes a vow makes an appointment with himself at some distant time or place. The danger of it is that himself should not keep the appointment. And in modern times this terror of one's self, of the weakness and mutability of one's self, has perilously increased, and is the real basis of the objection to vows of any kind. A modern man refrains from swearing to count the leaves on every third tree in Holland Walk, not because it is silly to do so (he does many sillier things), but because he has a profound conviction that before he had got to the three hundred and seventy-ninth leaf on the first tree he would be excessively tired of the subject and want to go home to tea. In other words, we fear that by that time he will be, in the common but hideously significant phrase, another man.

Making a vow gives an individual the liberty to constrain themselves. Modernity rejects this notion because it rejects obedience (even to yourself) and it rejects truth (even being honest with yourself). So, in the sense that no one will take your vow seriously, yes they're irrelevant today. But to the Christian, who does not live in accord with the age, vows are very important and they can help us to be obedient to the Church and to Christ.

Good vows will keep our wills as closely aligned to God's as possible.