Many power MOSFETs require a high gate voltage for high-current loads, to ensure that they are fully turned on. There are some with logic-level inputs, though. The data sheets can be misleading, they often give the gate voltage for 250 mA current on the front page, and you find that they need 12V for 5A, say.
It's a good idea to put a resistor to ground on the gate if a MOSFET is driven by an MCU output. MCU pins are usually inputs on reset, and this could cause the gate to float momentarily, perhaps turning the device on, until the program starts running. You won't damage the MCU output by connecting it directly to a MOSFET gate.
The BS170 and 2N7000 are roughly equivalent to the BJTs you mentioned. The Zetex ZVN4206ASTZ has a maximum drain current of 600 mA. I don't think that you will find a small MOSFET that can be driven from 3.3V, though.
The 12V and 6A is a good starting point. This tells me you need a mosfet with a max drain-source voltage capability greater than 12V so 20V would be a minimum criteria for this.
You want to switch 6A and you'll want it to do so with minimum volt-drop - just like a relay contact so you are looking for Rds(on) below (say) 0.1 ohms. This means at 6A it will develop a small voltage across the device of 0.6V (ohms law).
However, that will produce a power disippation of 6 x 6 x 0.1 W = 3.6W so if you are looking for a surface mount device you would prefer a lower disippation of maybe 0.5W max.
This means Rds(on) would be more like 0.014 ohms.
So far, your application needs a 20V transistor, capable of switching 6A with an on resistance no more than 0.014 ohms.
Vgs is "like" the coil voltage on a relay - it's how much voltage you need to apply to the coil to get it to switch BUT for a FET it's a linear thing and, if you don't apply enough voltage, the mosfet will not turn on properly - its on-resistance will be too high, it'll get warm under load and have a volt or two across it when you want a nice low resistance.
You then need to inspect the details of the spec to see how much you need to apply to guarantee the low on-resistance you want. A bit more on this further down.
The IRFZ44N has on the front page of the data sheet: -
Vdss = 55V, Rds(on) = 17.5 milli ohms and Id = 49A
It's not a surface mount device therefore a little more heat generated isn't going to matter too much (with a heatsink) so it'll do what you want it to do but I'd research a device with smaller Vds (say 20V) and you'll probably find one with a lot less than 10 milli ohms on resistance.
If you look at the electrical characteristics on page 2 you'll see that the 17.5 milli ohms on resistance requires a 10V drive voltage on the gate (3rd line down in the table). Less than this drive level and the on-resistance rises as would the heat produced.
At this point I can't decide for you any more but I think you might be looking for a device that will operate from logic levels. In which case the IRFZ44N won't do.
The STB36NF06L is a little higher with the on-resistance but the spec does suggest it will work from a 5V drive on the gate - see electrical characteristics (ON) but i'd still be tempted to find one that is more suitable.
I'd be tempted by this. The PH2520U is a 20V, 100A, 2.7 milli ohm device when the gate voltage is 4.5V. If your logic levels are 3V3 check figure 9 to see it will work well at 3V3.
One last thought about things - you are wanting to PWM a load and if the frequency is high you'll find that the gate capacitance takes some drive current into the gate to get it moving up and down quickly. Sometimes it better to trade off on-resistance to find a device with lower Vgs capacitance. You're into horse-trading now. Keep as low as you can on switching frequency and it should drive ok from a 5V logic pin.
Best Answer
In order to turn off, a MOSFET must have a \$V_{gs}\$ which is less than some threshold. Ideally it should be 0.
For the lower transistors (the NMOS ones), this is fairly easy to understand - the source is connected to ground, so the \$V_{gs}\$ is calculated simply as equal to the voltage applied to the gate. If you have say a 5V MCU, this would be either 5V (on) or 0V (off).
For the upper transistor (the PMOS ones), it is a bit more confusing. The PMOS transistors have a \$V_{gs}\$ which is negative to be turned on. To turn on the gate must be at a lower potential than the source. To turn off, the gate must be at a potential close enough to the source to be less than a threshold.
Because the source of the PMOS is tied to 12V in the first circuit, its gate must be around 12V in order to turn off. If you drive the gate with a \$5V\$ signal directly, its gate is going to be \$V_{gs} = V_{io} - V_{sup} = 5 - 12 = -7V\$. If you drive it with a \$0V\$ signal, then \$V_{gs} = V_{io} - V_{sup} = 0 - 12 = -12V\$. If you consider that the threshold would be somewhere in the region of \$-1.5V\$, clearly the PMOS would be on in both cases - you could never turn it off.
To get around this, a level shifter must be added such that the gate is driven with either \$12V\$ (off) or \$0V\$ (on). The BJT in the circuit acts as a level shifter to convert the 5V control signal into the required 12V signal for the gate of the PMOS.
However, there is a side effect of the BJT circuit used which is that it inverts the control signal - a 0 becomes \$12V\$ (off) and a 1 becomes \$0V\$ (on). As a result the control signals have to be connected diagonally because you never want both transistors on the same half-bridge to be on at the same time - otherwise you short out the power supply.
In your latter circuit your motor supply and control signals are at the same level which means a level shifter is not needed - the 5V control signal can turn off the PMOS because it can create \$V_{gs} = V_{io} - V_{sup} = 5 - 5 = 0V\$. Because you no longer have the inverting level shifter, the MOSFETs are connected so that both transistors in the same half-bridge have the same control signal. A 0 now turns on the PMOS but turns off the NMOS, and a 1 turns on the NMOS but turns off the PMOS. This ensures both transistors in the same half-bridge never turn on at the same time.