Flashme is a firmware mod for your DS. It doesn't replace your bios, only patches key areas to allow things like booting DS code from the GBA cart, and booting DS code via wifi. It can not be detected by nintendo, and you can still go online after flashing.

Pros of flashing:
1. If you get the DSbrick virus, you can recover your DS (not widespread).
2. Faster start up (no health warning screens)
3. Can run DS code from GBA slot with or without anything in DS slot (though sleep mode doesn't work)
4. Can play multiplayer games that check digital signature.
5. Sleep Mode with MML

Cons of flashing:
1. Can brick your DSlite if your not careful.
2. If the DS loses power while flashing and the flash is less than 25% complete, your DS is a brick. Between 25% and 60% it will appear like a brick but is recoverable, between 60-99% it should be reflashed though it might seem normal.

Differences in FlashMe versions:
V5 enables Wifi support in games. Previous versions will make the ds appear bricked, but it is recoverable.
V6 general bugfixes and support for direct booting into Neoflash's MK series flashcarts without a GBA cart.
V7 Dslite support.

Update: With slot1 flashcarts prevalent in the market now, there is much less reason to flash. Slot1 flashcarts typically will let you boot DS code from the GBA slot (built in passme) and do not themselves require a passme/flash. Everything above still applies, but flashing isn't as required with a slot 1 solution.