This is a description of a most basic Li-Ion charger:
If a Li-Ion battery is discharged, the charger starts the charging cycle at 4,2v at the maximum current the battery can consume. If you charger only delivers 10ma then it will take 10ma if it delvers 1000ma then the battery will take 1000ma. As the battery gets charged up the internal resistance of it goes up and draws less and less current until it doesn't draw any (or very little). Those numbers are recommended numbers. The only real difference is the time it takes to charge will be a little longer with 0.3C then 0.5C or 1C
Think of it like your old NiCads or NiMh batteries. The more current you pump in the faster they charge but they will still charge with 10ma but it takes overnight instead of a couple of hours.
I have built Li-Ion Chargers. You build a regulated voltage supply and set the max current. If you put an amp meter in series with the + terminal you will see that when fully discharged it will take all the current the supply can pump out enough to drop the voltage but as it charges the current keeps dropping and the voltege goes up until it reaches 4.2v from 2.7v which is discharged. The reason is that it draws all the current you can supply which drops the voltage until the battery gets up to the 4.2v then the current starts dropping. This is also why its important that if you could supply a lot of current to watch the temp of the battery, because if its shorted and you can supply enough current, it gets hot and it could catch fire and explode and they do. Although with low current its very very rare. The Sony recall was because some batteries in the pack could short which cause the above problem. Also remember that a laptop pack is several batteries in parallel which ups the current as well as series to up the voltage. SO you get high voltages at high current. This is why there is a circuit board inside a laptop battery, for safety shutdown and to regulate the charge. These little cell battery's don't have these. Were only dealing with one cell here.
What I have described is the most basic Li-Ion charger, what we have. A good Li-Ion charger is much more complex.
If you need more information do a google search. Also a good place to start is an Electric Model Airplane forum. Where their pushing the limits of Li-Poly and Ion batteries. This and the auto industry is driving the Li-Ion battery market. I use 7.4V at more than 2 amps for my small airplane. I sometimes get my batteries from old laptop packs. I look for hi current small batteries and wire them in series parallel to get the voltage and current. Li- Poly batteries are small pouches that can be squashed and molded, they are very soft and squishie. Unlike Li-Ion which is hard and stiff.
http://www.rcgroups.com/batteries-and-chargers-129/
Here is a partial write up.
Li-ion battery charging
Li-xx (Li-ion or Li-ion polymer) batteries are charged in three stages. The qualification and precharge stage verifies if the battery's temperature is between the limits recommended by the cell manufacturer, typically 10° to 45°C.
Charge is suspended until the cell reaches normal operating temperature. If the battery is deeply discharged (below 3.0 V), a precharge current in the range of 10% of the fast-charge current is applied that safely raises the battery voltage prior to the fast-charge stage. A safety timer helps to suspend charge if the battery voltage does not come up during a certain period of time. Charge qualification and precharge increase safety and cycle life of the battery.
In stage two, the battery is charged with a constant current of 1C or less, "C" being the rated capacity of the Li-xx cell. Once the battery has reached its specified voltage limit of 4.1 or 4.2 V, about 70% of the battery capacity is typically replenished.
Stage three charges the battery with a constant voltage. To maximize safety and capacity, this voltage needs to be regulated to ±1%. During the voltage regulation phase, the charge current tapers down, and in most cases charge is terminated when the charge current reaches about 10% of the fast-charge current level, or C/10. An internal charge timer might provide a backup safety for charge termination.