Check your laptop model online to see if it has a SATA SSD or M.2 SSD. Also can you share the desktop motherboard model.
As
@rabid.parrots pointed out there are 3 types of consumer grade SSDs-
1) SATA SSD.This is a squarish 2.5" device as shown in first screenshot which uses standard SATA and power connectors. This is the one to go for if you want to maintain maximum compatibility across desktop/laptops albeit its a tad slower than M.2 nvme.
2) M.2 SATA SSD- Performance is the same as a SATA SSD but its a RAM-esque circuit board (mostly 22x80 mm in size). Newer laptops use this type to cut down on space usage. However you can't use this on your desktop if you motherboard doesn't have a M.2 slot.Pricing is similar for for these two types of SSDs.
3)M.2 nvme- Form factor is the same as the M2. SATA SSD but this one uses PCIex4 connection instead of SATA connection and is much faster than either 1 and 2.
However you don't seem like a Tier 1 hardware enthusiast and so I would suggest going for options1/2 based on your laptop and mobo model as M.2 nvme is much faster in synthetic benchmarks but real world difference in daily usage (especially in games) is minimal.
I would suggest going for Samsung Evo line as I have been using only Samsung SSDs in my desktops since 2013(although my XPS13 has some other nvme SSD) and I have faced no issues at all.