In the previous mitzvah (and in Mitzvah #438), we discussed one's obligation to fulfill vows and voluntary offerings which he committed himself to bring. In this mitzvah, we see the obligation to fulfill other types of commitments a person might have taken upon himself. (Nachmanides limits this to religious obligations, such as donating a certain sum to charity.)
The difference between an obligation of this sort and one involving a sacrifice (as in the previous mitzvah) is that one has three Festivals to bring his sacrifice. If he commits to, say, give charity, that bill is due immediately.
The reason for this mitzvah is as in the previous one: if someone voluntarily pledges to do something out of gratitude to God, it's unseemly to then treat his commitment lightly. Unlike the case of the voluntary sacrifice, where the Torah does not require him to make a special trip to Jerusalem, there's no reason to delay paying his debt in this case. (For example, if he pledged money to charity, there's no shortage of worthy recipients to whom he might give it.)
This mitzvah applies in all times and places. It is discussed in the Talmud in tractate Rosh Hashana on page 6a and in the third chapter of tractate Shevuos. It is codified in the Shulchan Aruch in Yoreh Deah 204. This mitzvah is #94 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos and #39 of the 77 positive mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.