Step 1: Terminology
When you look at a LiPo's data sheet or casing, you will notice it has a lot of specs.
Cell arrangement - Described using the format xSyP (where x and y are integers), this tells you how the cells in the battery are wired up. Batteries are made up of cells, whose voltage is determined by cell chemistry and whose capacity is determined by energy density and physical size of the cell. S stands for series and P stands for parallel. As you may know, series adds the voltage of the cells and parallel adds the capacity of the cells, so a combination of cells in series and parallel results in a battery. The battery shown in the second image reads that it has an arrangement of 3S1P, meaning it has 3 cells that are all in series with no parallel wiring. This may seem confusing because it says "1P," but think of the arrangement as a grid. By multiplying the 3 and the 1, you get the total number of cells in the battery, which in this case is 3. If it were a 3S2P battery, there would be 2 sets of 3 series-wired cells in parallel, resulting in 6 cells total. Often times the parallel arrangement is omitted when discussing batteries, because most packs are 1P (so instead of saying you're using a 3S1P pack, you may as well just say 3S).
Capacity - Usually measured in mAh (milliamp hours), this is determined by the cell arrangement (parallel) and tells you how long you can expect the battery to last on a charge (although it's not quite that simple). 2600mAh as shown on the battery in the picture is equal to 2.6Ah (amp hours), a format you may be more familiar with on larger batteries, like the SLA (sealed lead acid) one in your car, which is probably around 50Ah. A capacity of 2600mAh means that the battery can discharge at 2.6 amps for one hour (hence "amp hours"), 1.3 amps for 2 hours, etc., before it runs out of "juice." Because the battery shown has a 1P arrangement, each cell has a capacity of 2600mAh.
Voltage -The voltage of a battery is also determined by the cell arrangement (series), and there are a few common voltage measurements worth noting:
Charged - the voltage of a fully-charged LiPo cell is 4.20V, and charging above this will damage the cell.
Nominal - this can be considered a sort of "half-charged" voltage, as it is 3.70V, in between charged and discharged. Nominal voltage is what manufacturers use when describing the voltage of their batteries.
Discharged - the voltage of a discharged LiPo cell is 3.00V, and discharging below this will definitely damage the cell.
Because the battery shown has a 3S arrangement, it is marked with its nominal voltage of 11.1V (3.70V*3 cells). A fully charged 3S pack is 12.60V and a fully discharged 3S pack is 9.00V.
Constant C Rating (Discharge) - The constant C rating (in relation to discharge) tells you how many amps can be safely drawn from the battery constantly. The "C" in a rating of xC (where x is an integer) actually stands for the capacity of the battery in Ah. By multiplying the C rating's coefficient by the capacity of the battery in Ah, you can determine the sort of amperage you can draw. In the case of this battery, with a capacity of 2600mAh (2.6Ah) and a C rating of 55C (that's pretty high, FYI), I can multiply 55*2.6 and get the max constant output of my battery, which is 143A.
Burst C Rating (Discharge) - In addition to the constant C rating, there is also a burst C rating, which is higher. Most of the time, the "burst" is rated for 10 seconds. Although it is not marked on the battery itself in the picture, it says in the documentation that this battery's 10 second burst rating is 80C. So, 80*2.6 is 208A burst. That's a lot! It's worth noting that your LiPo won't last long when that many amps are being drawn from it. At 208A, a 2600mAh LiPo will last approximately 45 seconds.
C Rating (Charge) - Determined in the same fashion as the C ratings for discharge, the C rating for charge tells you at what amperage you can safely charge your battery. This information is generally listed on the back of the battery with all the safety information. For the battery shown, it happens to be 5C, which means that it can be charged at 13A (2.6*5). We'll be talking a lot more about charge rates later...