Halloween is a holiday celebrated on the night of October 31, usually by children dressing in costumes and going door-to-door collecting candy.
"Originally a Celtic festival, Samhain, the last day of October became All Hallows Eve in the eighth century when Pope Gregory III moved All Saints' Day (or Hallows' Day) to November 1. The move was a means to claim the day for Christians, yet connections to pagan, Wiccan, and Druid beliefs remained." 
Although various earlier feasts and celebrations were observed to remember the departed saints, Gregory's Feast of All Saints was apparently the first such annual observance (and thus not "moved" from any other date). The Feast of All Souls (November 2) was later installed as a means of remembering the souls in Purgatory.
Some sources place the pagan celebration of Samhain later in November: "Although little is known about these celebrations, it seems that the Samhain festivities were observed between November 5-7 (midway between the autumnal equinox and the winter solstice) with a one-week series of events, ending with the feast of 'the dead,' which marked the beginning of the new Celtic year." 
It is unknown exactly how much influence the Celtic celebration had on the institution of the All Saints, but contemporary American practices almost certainly have a direct root through the large Irish (Celtic) immigrant population.
Halloween -- literally "All Hallow's Eve" -- is celebrated on the "eve" before All Saints ("All Hallowed").
Arguments for abstaining
"More than a thousand years ago Christians confronted pagan rites appeasing the lord of death and evil spirits. Halloween's unsavory beginnings preceded Christ's birth when the druids, in what is now Britain and France, observed the end of summer with sacrifices to the gods. It was the beginning of the Celtic year, and they believed Samhain, the lord of death, sent evil spirits abroad to attack humans, who could escape only by assuming disguises and looking like evil spirits themselves. The waning of the sun and the approach of dark winter made the evil spirits rejoice and play nasty tricks. Most of our Halloween practices can be traced back to the old pagan rites and superstitions." 
Emphasis on evil and the occult
"Witches, sorcerers, and ghosts are popular figures during Halloween. But the Bible strictly forbids occult practices such as witchcraft, sorcery, divination, interpreting omens, casting spells, and consulting a medium (Deut. 18:9-11). These counterfeit attempts to see the future, control one's fate, or obtain guidance are forbidden because God's people are to trust him for their future and seek his guidance." 
"No one can deny that Halloween is mainly focused on fear, darkness, monsters, the occult and violence. Are these the kinds of things that we should be exposing our children or ourselves to? ...Halloween is a holiday that glorifies that which is dark, that which is diabolical. " 
See Ephesians 5:11 and 1 Thessalonians 5:22
Misappropriation of money
Christians who are convicted not to participate in this holiday also believe that their money could be used in better ways. Rather than spend money on costumes, parties, candy, etc..., some Christians would rather give their money to the Church or to ministries instead.
Arguments for participating
Reinterpret or give new meaning
"For the children who will come to our door asking for candy, Halloween means something different from its original intent. Anthropologists talk about cultural reinterpretation, which is the process whereby an imported cultural trait or feature is reinterpreted to make it compatible with the values of the society embracing it. Presumably, the parents of little children who come to the door every year have reinterpreted Halloween, or ignored its original meaning so that their kids can have fun without the ugly elements usually associated with Halloween.
"Strictly speaking, it is impossible to reinterpret Halloween. Unlike Christmas and Easter, which also had pagan origins but were successfully changed into Christian holidays, Halloween is still pagan. Parents who dress their children in clown suits don't really reinterpret the holiday. They just divest it of meaning in order to turn the children loose for an evening of harmless fun." 
"[T]here is a place for some harmless fun. Kids love to dress up and pretend. If the Halloween experience is focused on fantasy rather than on the occult, I see no harm in it. Make costumes for your children that represent fun characters, such as Mickey Mouse or an elderly grandmother, and then let them go door-to-door asking for treats. This side of Halloween can be thoroughly enjoyable for little ones." James Dobson, 
Don't prevent kids from gathering all the free candy they can because of ancient "pagan" traditions. But, beware of the real danger: cavities. Adults, that doesn't mean pass out apples. Kids, that means brush your teeth before bed.
'Baptize' the holiday
"...celebrate the fact that at death we pass from the land of shadows into the land of light." 
Time with one's kids
Many parents look forward to Halloween as it allows them to spend time with their kids. Having the opportunity to set the focus of the evening, many parents pray with their kids before they leave to "trick or treat", and at the end of the night make a point, even as "childish" as it may seem, to give thanks to God for the candy that was freely given. Some have even taken the chance to use "free candy" to talk to their kids and relate this to God's grace that is given freely to those who believe.
Many parents who allow their kids to go door-to-door for candy also have their children carry pieces of paper with passages from the Bible. When the child receives their candy, in return they leave the person with a passage from the Bible concerning God's love, Jesus, salvation, etc.
Reformation Day is a day in remembrance of the Protestant Reformation. It takes place on October 31 and is official holiday in a lot of countries.
It has been encouraged to throw a party where each individual shows up wearing a costume. Even though kids may be wearing costumes that resemble withes, ghosts, or things of this sort, the focus of the party does not have to be "pagan" or "evil".
Remembrance of departed believers
"Remember your leaders, those who spoke to you the word of God. Consider the outcome of their way of life, and imitate their faith." -Hebrews 13:7
"There are other values in a meeting convened to remember those who have passed into heaven. Believers who have lost friends or family members could be encouraged to talk about them openly. What the Bible teaches about the state of the dead who die in the Lord could be reviewed, and the occasion could be a fine opportunity for proclaiming that Christ is Conqueror over death and hell and Satan." 
Trunk & Treat
Some churches host what is called a "Trunk and Treat". At this event, parents arrive having dressed up their cars, using their trunk (or truck bed) as the area where candy is passed out, decorations are done, or anything of this sort. Parents fill up the parking lot, and kids go "car to car" viewing each car's "costume" and getting candy. It provides a time for parents to allow their kids to enjoy themselves in a safe environment and all are able to enjoy fellowship with others from their church.
Cautions and ideas
Whether one allows their child to participate during Halloween, parents should be cautioned of a few things. If you decide not allow your child to participate, be aware of their reactions and feelings. Typically children (or even teens) will feel left out from social events, time with their friends, or will feel abnormal, different, or "not like the other kids". This is not a reason to go against a personal conviction, but it does mean to be conscious of their feelings, and to act as a support in their time of disappointment (not to say all kids will be disappointed). Whether your child is following Jesus or not, this is a great opportunity to bring your child to God's Word and introduce or remind them of the commitment to follow Jesus at any cost. Rent a movie, get some candy, or plan something fun for the evening.
For those who will be allowing their child to participate, be cautious of your child's motives to participate. Some may get into the "evil" side of Halloween too much, while others may want to go for the social interaction or other reasons. If your child is following Jesus, help remind them of their commitment to Christ, and to remain an example to their friends as they "trick or treat" (this is one of many examples). If you are unsure if your child is following Jesus, and will still be allowing them to go, perhaps take time to discuss with them how to make good choices. Find ways to relate this to Jesus, or God's Word.
- Suggested Guidelines for Christians by Richard Bucher
- Annie's "Halloween Welcome" Page, A Christian Perspective on the Holiday
- Halloween (Wikipedia)
- The History of Halloween (The History Channel)
- Halloween (The Dane)
- The Inevitable Halloween Discussion, by Tim Challies
- Home for Halloween, by John Fischer
- What to Do About Halloween, by James Dobson
- Halloween: As Viewed by Evangelical Christians, by B.A. Robinson
Halloween/All Saints' Day (collection of Christianity Today articles)
- Why I Let My Kids Go Trick-or-Treating, by Ellie Lofaro
- Is Halloween a Witches' Brew?, by Harold L. Myra
- In Perspective: The Christian and the Jack-o'-Lantern, by Todd Hertz
- Why Christians should embrace the "devilish" holiday with gusto—and laughter, by Anderson M. Rearick III
- The holy, hellish, hodgepodge history of Halloween, by James Harleman
- A Perspective on Halloween, by Randy Alcorn
- What are your thoughts on Halloween?, by John Piper
- Why All Good Christians Should Celebrate Halloween, by George Robinson
- The History of Halloween Revisited, by James Harleman
- Christians and Halloween, by Travis Allen (Grace To You)
- What Christians Should Know about Halloween, by Justin Holcomb
- Halloween, by Douglas Wilson
- The Holy, Horror, and Halloween, by Douglas Wilson
- Everything you always wanted to know about Satan, demons, and zombies but were afraid to ask, by Mark Driscoll
- An Open Letter About Halloween, by Jim Daly