The Broadcom datasheet linked from raspberrypi.org in Boštjan Jerko answer is a great reference; for example, section 6.2 describes the various functions that can be assigned to each pin as per joan's answer. To summarize the meaning of the ones indicated on your card:
I2C, SDA & SCL: In the Broadcom datasheet these are only mentioned directly in that section 6.2 table, but they are for an I2C bus, which is discussed in the section on the BSC (Broadcom serial controller). The 'DA' in SDA stands for data, the 'CL' in SCL stands for clock; the S stands for serial. You can do more reading about the significance of the clock line for various types of computer bus, but you do not have to understand it on that level to use it -- as joan says, there are programming libraries that abstract that away. You will probably find I2C devices that come with their own userspace drivers and the linux kernel includes some as well. Most computers have an I2C bus, presumably for some of the purposes listed by wikipedia, such as interfacing with the RTC (real time clock) and configuring memory. However, it is not exposed, meaning you can't attach anything else to it, and there are a lot of interesting things that could be attached -- pretty much any kind of common sensor (barometers, accelerometers, gyroscopes, luminometers, etc.) as well as output devices and displays. You can buy a USB to I2C adapter for a normal computer, but they cost a few hundred dollars. You can attach multiple devices to the exposed bus on the pi.
UART, TXD & RXD: This is a traditional serial line; for decades most computers have had a port for this and a port for parallel.1 Some pi oriented OS distros such as Raspbian by default boot with this serial line active as a console, and you can plug the other end into another computer and use some appropriate software to communicate with it. Note this interface does not have a clock line; the two pins may be used for full duplex communication (simultaneous transmit and receive).
PCM, CLK/DIN/DOUT/FS: PCM is is how uncompressed digital audio is encoded. The data stream is serial, but interpreting this correctly is best done with a separate clock line (more lowest level stuff).
SPI, MOSI/MISO/CE0/CE1: SPI is a serial bus protocol serving many of the same purposes as I2C, but because there are more wires, it can operate in full duplex which makes it faster and more flexible.
1. The pi actually does not have any particular style of parallel bus exposed, but presumably you could implement one with the some of the pins.