I need to backup my server from data center having around 700GB of data.
Which external drive shall I use (it will be directly connected to the server) for backup?
The one that meets your storage needs and budget.
How long it will take to copy data from server to external drive (there are around 100 files of total size 700GB)?
This depends on the device you purchase. A USB 2.0 external drive would take longer than a USB 3.0 external drive which would take longer then an eSATA external drive which would take (slightly) longer than an internal SATA drive.
According to the File Transfer Time Calculator, copying 700 GB of data over USB 3, will take just over 32 minutes.
But then, there are other factors to consider, such as what the physical media is - is it solid-state, or spinning disk? If it's a spinning disk, what speed (rpm) is it?
As for which type to buy - well, you have to think about what you're using it for. Is this a one-time backup of a remote server? Or is it going to be left plugged in and used to make ad-hoc or scheduled copies of data on the server? If it's a one-time backup, and you're planning to keep it for less than 10 years, you'll probably be fine with an SSD.
If you're regularly writing/wiping the backup data, you might want to look a little more closely at the failure rates and how many write cycles are supported on the drive. If you're filling the drive up every day with fresh data, and the drive is cheap and expected to fail after 2,000 cycles, you will end up losing the drive in just under 5½ years (assuming you use an operating system which supports the TRIM operation).
You also need to think about what filesystem you will be using. If this is a Windows environment, and you need NTFS, beware that you should disable write caching on the disk, else you might find the disk is actually empty and you have no backup at all.
Another point to consider is the requirement for an external power supply. If this is in a datacenter environment, you need to make sure that you have a free power socket for it, and that the power adapter fits the plugs in your rack. You also need to think about what happens when the power supply fails. Do you have another spare that can be used, and how long will it take you to swap it out?
