Ok, from grub type ls (hd0,1)/
you should see a file named vmlinuz or linux, and initrd.img

Type linux (hd0,1)/vmlinuz root=/dev/sda1 or linux (hd0,1)/linux root=/dev/sda1 depending on what you found with ls (hd0,1)/, then:

initrd (hd0,1)/initrd.img boot 

If you get initramfs rescue mode enter your password, then startx. You should now have a desktop.

Use gparted to check your file system, if it reports an error, then you need to boot from a LiveCD or other media to fix it …. DO NOT attempt to repair a mounted partition.

The following three commands fix many grub boot problems. They run quick so just do all three instead of trying to find which one you need.

sudo grub-install /dev/sda && sudo update-grub && sudo update-initramfs -u 

Reboot and see what you get.