Is it possible to use the galileo's ethernet port through arduino IDE? Or should I buy an arduino ethernet shield? Will I be able to use galileo as a server to which arduino board can send data to?
I suppose you mean the "Galileo integrated Ethernet port" (the on-board RJ45). Then the answer is "no, not directly". The whole code provided by Intel in Arduino 1.6 IDE applies to the W5100 based shield, not the integrated Ethernet interface (which is a bit stupid, imho...) Technically, however, this is possible, but you will need to create a library in C/C++, which interfaces with the system lib (the Linux one). You can then access to the wonderful world of BSD sockets (much more powerful than then sockets within the W5100) Note however that the configuration of the integrated Ethernet board is not very easy with the Galileo, because the default Yocto image does not allow you to change default IP parameters (the Yocto image uses DHCP by default). So technically, you need then to deal with a startup script which changes the IP parameters at startup, but it's pretty heavy to do.
The Galileo board is Arduino compatible; meaning that you can run Arduino sketches on it, and use its I/O pins the same way you would with, say, Arduino Uno.
Therefor you can communicate between Galileo and Arduino the same way you would with two Arduino devices. This can be done using your favorite communication channel - Serial, I2C, OneWire, RF or even Infrared - it all depends what exactly are your needs, knowledge and timing requirements. Of course, doing it this way, would require you to have code running on both the Galileo and the Arduino, in charge of the inter-communication.
By doing so, you could utilize the Galileo as a 'server' as you say, that will receive commands from the Arduino, send them over the network and reply to the Arduino with the response.
Yup - the Galileo Ethernet port works the same way as any other Arduino. For some sketch examples have a look at
File > Examples > Ethernet