FET Type: I'm not sure what the difference is between N and P channel
The internal construction of a mosfet is different and you need different voltage levels to switch it on. Higher than source for N channel and lower than source for P channel. As you will be switching 25V load from a 5V microcontroller, choose an N channel logic level mosfet.
Drain to Source Voltate (Vdss): I'm assuming this is the max voltage it can handle going through it, so I should be finding a MOSFET that will support 25 V+?
It's the maximum voltage whitch the mosfet can withstand without letting the current to run through it.
By the rule of thumb you should double the rating to get a reliably working system. So, look for a mosfet with Vds in the range of 50V-60V. It would be OK to use a 25V mosfet but you usually don't want to operate near maximum limited values.
Current - Continuous Drain (Id): Assuming this is the max amperage going through it, so looking for one with 12.5 A+
Again - double it.
Vgs(th) (Max): I think this has something to do with the activation voltage applied to the gate that will make it activate, so I need one with less than 5 V?
Yes, mosfet dissipates least power when it's either fully on or off. Look at the graphs in the datasheet that specify Rdson depending on Vg - you want Rdson as small as possible, so you want to drive the gate above the Vgth. But note, that there is a maximum value that can be safely applied to a gate - Vgsmax. You should be safe driving it with a microcontroller, just a point to note.
Power - Max: Assuming this is the max power it can handle. I've calculated the power the solenoid would need as P = V*I = 25 V * 12.5 A = 312.5 W, so I need a MOSFET that can handle more than 312.5 W?
No, power dissipated by a mosfet would be I*I*Rdson - that's why you want as little Rdson as possible.
I don't know what Rds On (Max), Gate Charge (Qg), or Input Capacitance (Ciss) mean. Are they important for my uses?
When a mosfet is on, it's not an ideal conductor with no resistance. Rdson is the resistance of the mosfet and is dependent on different factors, datasheets usually give graphs how Rdson changes with different parameters.
You don't have to deal with gate charge and input capacitance in you application as fast (submilisecond) switching is not required. A mosfet gate presents itself as a capacitor to a driving circuitry and as it takes time for a capacitor to charge, it takes time for a mosfet to turn on that's why in high speed applications special mosfet driver ics are used that force high currents into gate to charge this capacitance as quickly as possible.
You can find cheaper mosfets with lower Rdson, just use the parametric search on digikey. Pay attention to the graph that displays Rdson against Vgth - sometimes manufacturers claim 4V Vgth and 4mOhm Rdsn, but when you look at the graph you see, that at 4V it's 20mOhm and you need to get to 9V to get the advertised 4mOhm Rdson.
No, this transistor cannot be expected to do this job for long, if it can do it at all.
From the datasheet, look at the "On characteristics" on page 2.
First, its free air power rating is 0.625W, which means Vce had better be 1.25V or less at 500 ma.
Then, gain (hFE) is shown at different Vce voltages and currents. But significantly, not shown at Vce=1V and Ic=500mA, suggesting that the transistor is not rated to work under those conditions.
Finally, the CE saturation voltage is shown as 1.6V at 500mA, which exceeds the power rating shown above. You will be able to get away with that for a few seconds on a very low duty cycle. The MPS2222A would be a better choice, its Vce(sat) is shown as 1V here.
But...the above condition is achieved with Ib=50mA. This almost certainly exceeds the current available from your Arduino output pin.
If you are content to briefly overrate the device's power ratings, you could overcome the base current limitation using a second transistor as an emitter follower, to drive the base current you need.
(schematic editor isn't loading this morning, sorry)
Best Answer
The circuit is fine in theory.
Improvement in practice is required.
Adding a gate-source zener diode of say 12V (> Vgate_drive) is a very good idea indeed in all circuits with inductive load. This stops the gate being driven destructively high by "Miller capacitance" coupling to the drain during unexpected or extreme variations in drain voltage.
Mount the zener close to the MOSFET.
Connect Anode to source and Cathode to gate so that the zener does not usually conduct.
The 10k gate drive resistor (as shown) is large and will cause slow turn off and on and more power dissipation in the MOSFET. This is probably not a problem here.
The chosen MOSFET is very marginal in this application.
Far far far better MOSFETs available ex stock at Digikey include:
For 26c/10 Digikey IRLML6346 SOT23 pkg, 30V, 3.4A, 0.06 Ohm, Vgsth = 1.1V = gate threshold Voltage..
NDT3055 48c/10 TO251 leaded 60V, 12A, 0.1 Ohm, Vgsth = 2V
RFD14N05 71c/10 TO220 50V, 14A, 0.1 Ohm, 2V Vgsth.
ADDED
SUITABLE MOSFETS FOR 3V GATE DRIVE:
System just trashed my longer answer :-(. So - MOSFET MUST have Vth (threshold voltage) of no more than 2V to work properly with 3V3 supply controllers.
None of the suggested FETS meet this requirement.
They may work after a fashion on the present load but are underdriven and overly lossy and the solution does not extend well to larger loads.
It seems that IRF FETS in size range concerned that have Vth (of Vgsth) <= 2 volts ALL have 4 digit numerical codes starting with 7 except IRF3708.
OK FETs include IRFxxxx where xxxx = 3708 6607 7201 6321 7326 7342 7353 7403 7406 7416 7455 7463 7468 7470
There will be others but all the ones suggested seem to have Vth = 4V or 5V and are marginal or worse in this application.
Vgsth or Vth needs to be at least one Volt less and ideally several volts less than actual gate drive voltage.