The primary job of the oil you put into your engine is to stop the various metal surfaces from grinding together, causing rapid wear of the parts. At the same time it has to dissipate the heat generated from friction, to transfer part of the heat of combustion away, to hold the byproducts of the combustion in suspension, not allowing them to stick to the engine parts. The oil has to comply with all these requirements under significant pressure and a wide range of temperatures ranging from the chilled engine in a winter morning to the high temperatures in an operating engine. In addition, it has to retain its performance under these varying circumstances and to remain reasonably stable until the end of the recommended oil change period.
The most significant physical property of an engine oil, viscosity, describes the resistance of the oil to flow. Viscosity decreases as temperature rise. In the engine, the heat produced by the combustion and internal friction results in a thinner (less viscous) fluid. When the film between the mating surfaces is thin, the lubrication is not satisfactory. As the viscosity of the oil decreases because of the rising temperature, the lubricant will be less able to withstand the load, this generates more heat which, in turn, reduce the viscosity even further. Thick films, in contrast, give a stable lubrication: as the temperature rises, the viscosity drops together with the coefficient of friction between the sufaces. This reduces the heat which, in turn, rise the viscosity slightly. Like a feedback, this process stabilizes the thick film between the mating surfaces.
All oils respond with decreasing viscosity to increasing temperature, however, the amount of viscosity change due to a given temperature change depends heavily on the chemical composition of the liquid. The property of resisting changes in viscosity as the temperature changes is called viscosity index (VI); the higher this index, the better the oil resists viscosity changes. A lubricating oil with a low viscosity index would be detrimental in an engine: at low temperatures (cold cranking) the oil would be too viscous for proper circulation while at high temperatures (normal operation) it would thin out to the extent of letting the oil film between the mating metal surfaces break, resulting in severe wear.
Conventional (mineral) oils are mixed from different compounds (fractions) separated from crude oil by progressive boiling and distillation. Low viscosity fractions generally have lower boiling points. When mixed and used in the engine, they retain their boiling point meaning that at higher temperatures in the engine the low viscosity fractions will vaporize, leaving behind the high viscosity components. This prevents the oil from thinning too much as it warms up.
The engine oils contain many other additives such as anti-wear agents, extreme pressure agents (helping the oil hold up between surfaces with high contact stress), anti-rust agents, corrosion inhibitors, detergents and dispersants (to remove dirt and sludge as well as to hold them in suspension), friction modifiers, pour point depressants (to inhibit wax crystal growth at low temperatures; this is important for good cold cranking performance), viscosity index improvers (to rise the VI; these improvers wear out causing both the viscosity index to drop and the improver itself to remain in the oil, to be held in suspension), seal swell agents, anti-foam agents, antioxidants and metal deactivators.
Synthetic oils are manufactured by reacting various organic chemicals together. The basic components are thermally more stable. To achieve the same viscosity index, synthetics require considerably less improver. The base stocks used in synthetic oils have lower pour points, so there's little or no need for pour point depressants.
Contrary to popular belief, synthetic oils are not better than mineral ones as far as their specification is concerned. The advantage of synthetics is that they retain nearly the same quality while they are in your engine, up to the minute of the next oil change. They also can deliver the same performance in a much wider temperature range and they have a much lower tendency to form deposits. The minerals, on the other hand, start to wear out quickly as the improvers wear out and the lower viscosity fractions with lower boiling point start to boil off or oxidize. As a consequence, you should stick to synthetic oil not because of higher but more consistent performance.
As far as the oil manufacturer is concerned, stick to quality brand names. Citroën prefers Total but Castrol, BP, Shell, Aral, Valvoline, Agip or Amsoil are also excellent brands. Synthetics are more expensive than minerals. Their higher production costs make them impractical competitors in the lower product range, hence, synthetics are generally produced to meet the elevated requirements of higher quality engine oils, giving a higher price to start with. The more stable composition of synthetic oils makes it possible to change them less frequently than it is recommended with mineral oils, compensating for the price difference—although this is not necessarily wise, the oil itself might stay stable but the contamination resulting from use will still be present and might warrant the oil change at the recommended intervals.
If you (or a previous user) have been using mineral oil in the engine for years, you can still switch to synthetic without too much trouble, although some people mention that the new oil might dislodge old deposits of the mineral oil, causing minor leaks. Using engine flushing oils can aggravate this problem so it is best avoiding them; just change the oil and the filter.