In what order to children's teeth come in?

It really depends. Not every baby follows the exact same schedule. But, most babies get the lower central incisors (two bottom front teeth) first & then the upper central incisors after that. Beyond that, the sequence may be irregular so don't worry about that. It is not terribly important when the teeth come in, but more important your dental hygeine regimen. Brush the teeth twice a day & avoid bottles in bed.
Pattern varies, but. Eruption pattern may vary in any given case; usually lowers before uppers and front before back. The most important issue is symmetry - whatever the right side does the left should and vice versa. Deviation from that needs professional attention.
Front teeth first. Expect the lower front teeth to erupt between 6 and 10 month followed by upper front teeth , first molars will come at about 13 to 19 month, the canines between 16 and 22 month sand last molars at 2 to 2 1/2 year. At the age of 6, you need to watch for the first permanent molars to show up, not to be confused with a primary or baby tooth. Hope this help.
From 6 to 30 months. Baby teeth can start appearing at 6 months, but some babies don't have teeth until 1 year of age. The central incisors come first, usually lower before upper, and appear at 6-8 months. Lateral incisors appear at 9-10 months. First molars and canines appear at about 18 months, and second molars at 24-30 months; totaling 20 teeth in all. A dentist can evaluate any oddly shaped or positioned teeth.
Typical pattern is. Lower centrals, upper centrals, upper front sides, lower front sides, lower first molars, upper first molars, upper canines, lower canines, lower 2nd molars, upper 2nd molars. Typically starts at 6 mo. and finishes between 30-36 month. Variation from this pattern can be normal, but if a molar shows up at 4 mo. an no other teeth are in the mouth I would be suspicious. .
All Over the Place. I have seen kids with teeth as early as 3 months and as late as 16 months for the 1st teeth.Although lower central incisors are common as 1st, i think I have seen every pattern you can imagine.So it is highly variable.
Tooth eruption. Usually the lower front incisors should appear about 4-6 months after birth.
Varies. Normal order of eruption only occurs 43% of the time. 57% don't follow "normal" order as discussed in other answers to this question. Of greater importance than "order" is position. If teeth erupt crooked, or out of position, see an orthodontist immediately to reduce time and expense of ortho rx.