At the moment you were born, each celestial body (the sun, the moon, and the nine planets) was in a specific point in the sky. Your birth chart will tell you which Zodiac sign corresponds with each of those points. This is how you determine your planetary signs — your Venus sign, your Mars sign, and so forth.
A sign's ruling, or natural, planet imbues that sign with some of the traits we normally associate with it. The sun, for example, gives Leos their signature warmth and big personality. Likewise, we have Mercury, the planet of communication, to thank for our Gemini friends' chattiness and inquisitive nature.
Planetary placements that aren't of the "natural" sort are often interpreted as that planet's area of influence being expressed through a sign's specific personality: Someone with Mercury in Pisces will communicate differently than someone with this planet in a more outgoing sign, like Sagittarius, for instance. But, when that placement is natural, say, when someone has Mercury in Gemini, it's more accurate to say that Mercury's energy is being amplified by the sign, rather than filtered or distilled — thus, that aforementioned chattiness will be on full display with someone whose Mercury sign is Gemini.
Here's another example: The natural placement of Mars in Aries makes for a fiery combo. This planet governs our tempers, sexual appetites, and motivations, while Aries is best known for its passionate, look-before-you-leap approach to life and, to put it lightly, a slight tendency to be competitive. If you have this pairing in your chart, you probably act instinctively and impulsively. You hate when people aren't direct with you and you love a challenge. The qualities that Mars and Aries possess, respectively, are working in tandem in your chart to make you a live wire, probably what some might call hot-headed.
Don't feel like you're missing out if you don't have any natural placements at all. The planets play an important role in everyone's astrological identity — no matter where they appear in the chart. But, if you've always wondered why having Saturn in Capricorn helps make you a great leader (or, for that matter, what having Venus in Taurus has to do with your seriously flirtatious streak), now you know.