Power dissipation often comes in handy in transistor selection also.

P=IV=I^2*R = V^2/R

For MOSFETs, current flow & Rds is usually known, so P=I^2 * R is good

(and if you current & Rds, V=IR so you can determine Vds also)

For BJTs, current flow & Vce-sat is usually known, so P=IV is good

Vce-sat of ~0.7V vs Rds of 0.035ohm shows why MOSFETs are preferred for high current loads:

example: 1A & 0.7V = 700mW, while 1A & 0.035ohm = 35mW

typically ignored is the power from the base current - say for an NPN 15mA was used, and Vbe is 0.7V -that's another 10.5mW that is dissipated. 1.5% of the total from the example above, so ignoring it is fairly safe. At Arduino limits, say 35mA, max would be 0.035A * 0.7V = 24.5mW, still a fairly small amount.

For resistors, voltage & resistor value is known, so P=V^2/R is good.

For an LED for example: (Vs - Vf-led - Vtransistor)/current = resistor, or (Vs - Vf-led - Vtransistor)/ resistor = current, depending on what is being solved for (what resistor do I need for 20mA? if I use this 220 ohm resistor, what current will I get?)

Vtransistor is Vce-sat for BJTs, and current * Rds for MOSFETs

How is this applicable? Say you're wondering how much can you do with a 1/8w resistor,as you notice a warm smell coming from your circuit? 0.125W = I^2 * R, so with 20mA (max continuous for most LEDs), smallest R you can get by with is 0.125/(0.02^2) = 312 ohm, so a standard 330 would be safe.

That same 312 ohm resistor with a 1/4W rating could handle more current: P=I^2R, or Sqrt(P/R) = I, so Sqrt(0.25/312) = 28mA.

All things to consider as you select components.