First, u need to download the Linux Mint 17.3 iso file(= 1.4GB in size) to yr cptr, eg save to the "Download" folder for later access. Then u need to download/save n use Imgburn or Universal USB Installer to create a Live LM 17.3 DVD or USB-stick from the iso file.
.......From yr cptr's Boot Menu selection, boot up the Live LM 17.3 DVD or USB-stick. Connect the external/USB SSD to yr cptr. Click "Install Linux" on the Live LM 17.3 desktop n follow the prompts, eg click "Yes" to unmount the external SSD n "Erase disk n install LM 17.3 on the disk".
Fyi, it can be a problem installing Linux Mint 17.3 on "newer"(post-2009) SATA HDD that r 250GB or above in capacity or on SSD bc of newer HDD technology = partitions not properly aligned with the disk. Something to do with 512 bytes data blocks in older IDE/PATA HDD( that came in 120GB or less sizes) n 4096 bytes data blocks in newer HDD/SSD. Eg the error message during installation = "... offset of 3584 bytes from minimum alignment ..." .
If u get this error message during install of Linux Mint 17.3 on an external/USB HDD/SSD, u need to first use GParted(which is included in the Live DVD or USB-stick) to realign the modern SATA hard disk, ie unmount the HDD/SSD(usually is /dev/sdb or /dev/sdc), delete the whole old partition n create the 1st new partition meant for the "/" or root partition(eg 20GB in size), with 1 MiB(= the default value) at the "beginning of this space" n 0 MiB at the "end of this space", ensure "MiB" alignment box is checked(= do not select "cylinder"), set it as Primary partition n format to fat32 file system(= quicker process, ie no need to use ext 4 yet).
....... When creating the new 2nd partition meant for the home partition(eg 30GB in size), set 0 MiB for both the beginning n end of this space, Primary partition n format to fat32.
....... If u hv less than 4GB of RAM, u can also create a new 3rd partition for swap area(eg set as 2GB in size) for virtual memory/RAM on the hard disk, ... do as for the home partition above.
U can leave the remaining space as unallocated or free space. After the install n boot, u can use GParted to create a 4th new partition to store yr back-ups, movies, music, photos, files or install another Linux distro(= need to "sudo update-grub" at the Terminal to dual-boot).
Only after partitioning the external HDD with GParted, can u proceed with the Install via the Live DVD or USB-stick. Click "Something else", click the 1st partition that u had created earlier with GParted, click "Change", click "ext 4 file system", click "/" for Mount point, n so on, n also for the /home n swap partitions.
....... Ensure that the device for the bootloader installation is set to the external HDD/SSD, eg /dev/sdb or /dev/sdc. The cptr's internal HDD/SSD is usually identified as /dev/sda.
Seems, the Linux installer does not realign modern SATA HDD/SSD when u create a "New Partition Table" for the "/"(root), /home n swap partitions during the install process.
I hv just successfully installed n booted LM 17.3 on a "modern" SATA 250GB external/USB HDD which initially had the above error message about the 1st root partition not properly aligned with the HDD during install = could not proceed with the install. This was after many hours of reinstalling, trial n error.
....... Previously, I had no problem installing n booting LM 17.3 on an old external IDE/PATA 80GB HDD, ie no need to use GParted.