If you add batteries in series their voltage will add. If you add batteries in parallel their capacity (given in mAh most of the time) will add and the voltage will stay the same.
If you do a series and a parallel connection, the rules still stay the same, you could imagine to replace the parallel connection as a single battery with doubled capacity and then have just a series connection of two batteries again.
Note that this does not concern the current of the battery, which is still determined by Ohm's law.
So given that, both of your circuits are sort of equivalent. Just that the lamps in series will be both dark if one fails.
Batteries are capable of handling a certain current, limited by internal resistance and chemistry. This is also increased if you put them in parallel. The current capability is often given as a C-Rate which connects it to the capacity of the cell. 1C means it can handle a current equivalent to the capacity for 1 hour. So a 300mAh cell with 1C can handle a current of 300mA. Now if you put two of those in parallel, the C-rate stays the same 1C, but you increased the capacity to 600mAh, and now you can safely draw 600mA.
Going above the C-rate will reduce the energy you get out of the battery. So instead of 300mAh the capacity might be reduced to 200mAh if you draw a current of 500mA out of a single cell (there is no formula to this, it's specific for every cell). For rechargeable batteries, increasing the current draw and charging current reduces the lifetime of the batteries.
In series connection you have to be weary of the single cells drifting apart over time, which can lead to failures. This is especially a concern with lithium chemistries and rechargeable systems. As every cell is different (has a different capacity) some cells will be empty sooner than others and will be further discharged, below a critical point where the battery gets damaged. Same is true for charging, the cells with lower capacity will be charged to full faster, and after that will be overcharged. Depending on the chemistry different things can happen, lithium tends to be very dangerous in these scenarios, other chemistries are more forgiving and convert further charging current into heat.
A good online resource for all these things is the Battery University which I'd recommend to read first.
Start with DC batteries. Easier to understand. We'll turn them into vectors based on polarities which will help when we go to AC.
Series: \$E_T\ = E_1∡0°\ +\ E_2∡0°\$. If the batteries are identical. \$E_T\ = 2\ E_1∡0°\$. Current will be Ohm's Law, \$I\ =\ E_T\ /\ R\$. Twice the single battery current if R is constant.
Series: \$E_T\ = E_1∡0°\ +\ E_2∡180°\ =\ 0\$. Opposite polarities means no voltage or current.
Parallel: \$E_T\ = E_1∡0°\ =\ E_2∡0°\$. If the batteries are identical (same voltage, capacity, etc.) then \$I\ =\ E_T\ /\ R\$ and each battery will supply half the current to load. If the batteries are not identical in every way, current will flow from higher to lower and quickly discharge (in secondary cells).
Parallel: \$E_T\ = E_1∡0°\ =\ E_2∡180°\$. Both batteries act as loads and will quickly discharge and possibly explode or cause a fire. (Hence a lack of response to your question).
So single-phase AC.
Series: Each AC source has a magnitude and an phase angle \$V_1∡0°\$ and \$V_2∡θ\$. Now you must do vector addition on the two voltage sources to find the resultant. \$Vector\ V_R\ =\ Vector\ V_1\ +\ Vector\ V_2\$. Again Ohm's Law gives us current \$I\ =\ V_R\ /\ Z\ =\ V_R\ /\ R\$ - Assuming a resistive load. Current is also a vector. As phase angle between sources varies between 0° and 180°, current will vary from twice to 0.
Parallel: Now, assuming they are identical (same voltage, frequency, phase angle, etc.), current will flow from each source to load with each source providing half of the total.
If they are not identical, large currents will flow in the windings of the generator coils, hopefully activating protection or burning out generators.
In principle, if the main characteristics of devices are identical, you can generally parallel many electrical devices (drivers, regulators, transformers, generators, batteries) to get the same voltage and supply a proportional amount of current.
Three-phase generators can be paralleled if phase sequence, voltage levels and frequencies are the same. Generators can have different kW or kVA ratings with each supplying the same proportion of the load. 200kVA and 100kVA generators operating in parallel at 60%, would supply 120kVA and 60kVA.
A three-phase generators can be connected in wye or delta. When connecting a new generator in delta, odds are that the manufacturers instructions will state connect phase 1 to phase 2, phase 2 to phase 3. Then apply a voltmeter to unconnected terminals of 1 and 3 and measure the voltage when the generator is powered up. If 0V, phase 3 can be connected to phase 1. If one of the phases is backwards, \$2\ ×\ V_{PHASE}\$ would be applied to small impedance of series connected generators, creating large currents and quickly burning out coils.
Best Answer
If you connected the 1V @ 100mA power supply in SERIES with the 2V @ 200mA power supply, you would have a total of 3 volts. However the current is limited by the power supply with the least capacity, so you could expect only 100mA.
OTOH, connecting two power supplies of DIFFERENT voltages in parallel is an extremely bad idea. The two power supplies will fight with each other and may actually kill each other. NEVER RECOMMENDED.
Even connecting two power supplies of the SAME voltage together is not typically recommended because here in the Real World, they are never EXACTLY the same voltage. Ways around this include using diodes to isolate the power supplies from each other. Or using low-value resistors to provide a "buffer" between each supply and the load.