Oils come in various viscosities, also referred to as "weights" represented by the letter w. Viscosity describes how runny a liquid is, for example tomato ketchup is thick and less runny than water. The lower the number, the less viscous (more runny) the liquid.
In the past, engine oils only came in fixed weights, and in the winter a heavy weight oil would become even thicker when cold. Too thick to start the engine from cold, so a lighter grade would be used. In summer the situation reversed: the light weight winter oil would be even runnier in the summer heat. It would be too thin to protect an engine from serious wear once the engine reached operating temperatures and possibly too thin to protect the moving parts when the engine had it's first cold start of the day.
As a result, multigrade oils were developed. Instead of having 20w winter oil and 40w summer oil, chemists invented 20w40 oil. When the engine is cold, the oil behaves like a relatively thin, runny, low weight, high viscosity 20w oil. As the engine heats up, the viscosity actually increases to a maximum of 40w which gives better engine protection. The effect doesn't last forever though, as the oil suffers mechanical, chemical and heat stresses, the multigrade effect gradually weakens and gives less protection; the oil effectively becomes a lower weight oil. Synthetic oils have a much greater ability to avoid this degradation and remain fully effective for far longer, but when they do eventually degrade the performance drops off very suddenly. They do cost a lot more though: semi synthetic oils offer most of the benefits of fully synthetic oils but at substantially lower costs, but where a fully synth oil may have two or three times the lifespan of mineral oils, a semi synth will need changing as regularly as mineral oil. Unlike a mineral oil, it's likely to still be offering near optimal protection when changed whereas a mineral oil will have degraded significantly.
In short, if your manual states 20w40 oil, then any brand name oil (and most non-branded) which states 20w40 on the can will be fine as long as you stick to the oil change schedule. If you stretch the changes out a bit, or if you tend to give the engine hard use then go for the semi-synth. Unless you actually use the engine for full-on racing, then fully synth offers you no real benefits and costs you more. If you live in a hotter or cooler climate than the manufacturer intends, then they can advise you what other oils to use.