I installed Arch today and when I logged in I noticed that my filesystem is read only.
How can I fix this?
I would try to figure out why the boot process or kernel marked the file system as read-only.
Normally if something is detected during boot it will try to fix the filesystem with fsck and if it can't then not mount the filesystem.
I have seen running Linux servers switch their filesystems to read-only when there was a problem with the underlying SAN and the kernel couldn't write to the SAN.
As behrooz said, it might be something as simple - bug highly unlikely - as something changed your /etc/fstab file so that your filesystems are read-only.