Method 1:

On the latest kernels the SSD disks are automatically detected. This is how you verify:

Replace sda with your hard drive path.

$ cat /sys/block/sda/queue/rotational

You will a number as an output

1  means you have a HDD
0 means you have a SSD

Method 2:

This method needs root access and internet.

If you do not have the smartctl tool, you have to install it, like this:

$ sudo apt-get install smartmontools

Find out if you have a SSD or a normal HDD:

$ sudo smartctl -a /dev/sda

If you get this output, it means you have an SSD, else, it is an HDD:

Rotation Rate: Solid State Device