ninja2:
The first line in first table of the datasheet states the maximum current available at 25C is 9.2 A if you have 5V Gate to Source.
In the second table it states the "Gate Threshhold Voltage" is between 1V (min) and 2V (max), meaning it will start to be ON once more than 2V is applied G to S.
OK - that much I understand.
If you look closely at Fig 1 there's a little table in top left that shows each line in the graph is for a different VGS from 2.25 to 7.5V. So picking the line VGS = 3.5V it says that if Drain to Source volts is 10V (10 with power 1) the current will be about 10 to the power 0.2 or about 1.6 Amps.
This I don't. If I look at Fig 1, for Vgs of 3.5V (fourth line from the top), Drain to Source voltage of 10V (10^1), I see 10^? - ok, I understand that the exponent is between 0 and 1; that line sits almost on top of the 5th line above the 10^0 line; the 10^1 line is the 9th line (why not 10 divisions?) - what is the amount of the divisions spacing? How do you know this? How do you know that the line that it is sitting on is "0.2"? What value is the line above it? Below it? I can read this graph, but it seems like information is missing - or has to be inferred in some manner? So how?
Let's start at 10^0; ok - next line up from that (closer to 10^1) is...? There are 9 subdivisions, so it should be 10^0.11, and the next 10^0.22...until 10^0.99999 (ie, for all intents, 10^1)...right? So, that line is actually sitting on the 10^0.5555 line, or about 3.6 amps...not 1.6 amps...???
(it's bit hard to read the logarithmic graph, but just remember 10 to the power 0.1=1/10, 10 to the power 0 = 1, power 1 i= 10, power 2 = 100. For fractions between use your calculator or the POWER function in Excel)
I'm 37 years old - I understand exponents. 
But notice if VGS is 4V this rises to nearly 10A (which seems to contradict the 9.2A max!)
Anyway I'd say there's little doubt the Arduino output could drive this, it just comes down to setting your circuit up correctly.
Ok. Well, it seems aside from the above confusion I have on the graph, that I -am- understanding things somewhat properly...?
I find Wikipedia expains things really well ... have a look at MOSFET - Wikipedia
I'll do that...thank you. 
AlphaZeta:
What I was trying to say was that 5V gate voltage would not fully put the MOSFET in the switching mode. As ninja2 pointed out, the output of an Arduino may well be able to be high enough to be able to drive the motor, but it is not high enough to put the MOSFET in the saturation region (the flat top region). The saturation region shifts to the right as Vds goes up.
Ok - I can see that; and kinda implied I understood that for what I was writing about - but perhaps not for the OP's application...
For small signal applications, this is not an issue but to drive a motor you really want to get the point of operation to as far right as possible (i.e. higher Vgs) in order to obtain the lowest on-resistance and thus reduce the power dissipation of the MOSFET.
Right - and as I was stating before, the graphs seemed to indicate that anything greater than approximately 10V drain-to-source (to an extent) would work OK @ 5V Vgs...
The main difference between Logic MOSFET and normal MOSFET is that, for any given Vds the Vgs required to drive the device into the saturation region is lower. And particularly, if you are using a TTL device, it is guaranteed to turn on when Vgs is greater than 2.2V (which is the minimum TTL high for 5V supply), but like all the MOSFET, the required gate voltage goes up as the drain-to-source voltage goes up so for the operations Pokey desired (24V motor), a separate amplification state would be required (or using an optocoupler as I suggested before).
I think I see what you are saying; because both figure 1 and 2 show that in order to stay in the "flat region" as your Vds goes up, you have to keep increasing your Vgs - which is something I didn't realize before. However, it doesn't seem like I was too far off the mark for the voltages I was talking about (ie, 5V Vgs, 10V Vds @ approx 10A drain current - for the IRL520).
Would you agree with ninja2 that wikipedia is the best source for "information" in a general manner on MOSFETs, or do you have another source you would like to recommend?
Thank you. 