Connecting arduino to device

Is it possible to connect arduino to another device via USB/serial (FTDI) without PC?
I want to connect my arduino to the Fluke multimeter. I can connect Fluke to a PC and use HyperTerminal to communicate. Windows recognizes Fluke as USB serial device (like arduino).

I also tryed to connect two arduinos via USB cable and it doesn't work too. TX and RX LEDs aren't blinking.

Arduino by default is a USB slave device and not USB Host like your PC.
So you can not connect two arduino back to back with USB, it don't work.
You need Arduino USB host shield to connect to another device via USB.
http://blog.makezine.com/archive/2009/08/usb_host_shield_for_arduino.html

TX and RX LEDs aren't blinking.

these LEDs are not a function of the arduino. They are just lights that are switched on and off by the PCs driver software.

Is it possible to connect arduino to another device via USB/serial (FTDI) without PC?

USB, no. Serial, yes.

Some versions of the Arduino (including all the ones I own) don't use the FTDI chip: they connect the serial either directly or through a TTL-to-RS232 chip like the MAX232.

Does your meter have a direct USB interface? Or does it have a serial that requires a USB-to-serial cable?