The 5870 has 4 slots, at least the normal one does, the Eyefinity 6 edition has 6 slots.
The regular 5870 has 2 DVI ports, one HDMI port, and one DisplayPort port. As the 5870 chip only has 2 hardware DVI/VGA/HDMI signal generator units, you can only use 2 DVI/VGA/HDMI monitors at once. The third monitor absolutely has to be a DisplayPort monitor. Since DisplayPort monitors are rare and expensive, the more common approach is to use an active DisplayPort to VGA or active DisplayPort to Dual-Link DVI adapter. Note that you CANNOT USE PASSIVE ADAPTERS!!!!! This cannot be stressed enough, as most adapters you see (DisplayPort to DVI or HDMI) are passive. The passive adapters make the DisplayPort output DVI signals, but since the 5870 can only generate two DVI signals at once, you'll have to turn off one of the other monitors.
DisplayPort does NOT have a VGA compatibility mode, so any DisplayPort to VGA adapter will be active (this means it has a converter chip in the adapter that takes DisplayPort signals and converts them to VGA, not using a DVI or HDMI compatibility signal). This means that DisplayPort to VGA adapters are a sure-fire option for Eyefinity. The other kind are DisplayPort to Dual-Link DVI, these are typically larger boxes that have a USB connector to get additional power and they cost upwards of $100.
My setup is 3 Dell st2210 monitors (1920x1080), the left and center are hooked with DVI to the two DVI ports and the right monitor is hooked up with a Sapphire brand DisplayPort to VGA adapter. It works perfectly.
Here is the monitors plugged into my 5870: