The names of connected drives ( /dev/sda, /dev/sdb etc ) depend on the specific connectors where each drive is attached.
The drive names don't contain numbers. Numbered names (/dev/sda1, /dev/sda2 etc ) are partition names in each drive. In your case, I guess that your new drive is /dev/sdb and the extended partition in it is /dev/sdb1. The logical partition inside is /sdb5. If you observe carefully the graphic screen, you will see two separate frames of (slightly) different colors for the extended and logical partitions.
msdos partition type is the legacy partition type used since 30-35 years. It supports drives no bigger than 2 TiB. In case of bigger drives, only 2 TiB are visible by the operating system. It allows up to 4 primary partitions, or 3 primary + 1 extended (where you can create many logical ones). Linux uses numbers 1-4 for the 4 primary (or 3 primary + 1 extended) partitions. Logical partitions are always numbered 5, 6 and so on.
For bigger drives you have to use GPT instead of msdos partition type, in order to be able to access the entire drive's capacity.
To mount your new hard drive after reboot, you have to create a new entry in the /mnt directory and then use the "mount" command. Furthermore, you can configure mount parameters in the operating system to auto-mount it after each reboot. For this you can look for instructions concerning the specific o.s. you use, in this case Ubuntu 18.04. The first link I found is this one. Please look for the section "C. Mounting (including auto mount after reboot)".
Concerning GParted, please know that GParted never auto-mounts partitions, as this would prevent it from doing modifications needed for the partitioning operations. Under GPartetd you have always to manually mount any partition you want to access (you need to unmount it before proceeding to any modify operations).
Edited by mfleetwo:
Correct MSDOS partition table limits. 2 TiB, not 2 GiB.
*** It is highly recommended to backup any important files before doing resize/move operations. ***