The example you gave seems a bit challenging to make continuous tiles to (in the y axis), even with the offset tools in Photoshop and Gimp.
If you want to search for ready-made textures or tutorials on how to do this, use the keyword “seamless”.
For “intentionally repetitive” patterns like those you would use as wallpaper, carpeting, flooring, etc., you should be all set by using the plugins or the offset feature of your imaging applications.
For stuff like food, rocks, clouds, etc. (stuff placed in semi-random locations that still look evenly distributed), be careful of unwanted repetitive patterns as they would make your scene look fake. Although you will not have that hard line (seam), our eyes will still detect the repetition. Having a “large enough” image to seamlessly tile will help.