I have been given these dimensions: 115 3/8”w x 38”h for a print banner but I don't quite understand what it means! i.e: what is the width & height of the canvas I should work on?
Well you might not want to use Photoshop but regardless that means 115 inches + 3/8's of an inch is the width, and 38 inches is the height.
As far as if that's the Trim or whatnot, you'd have to ask printer.
If you are using Photoshop you just use a calculator to divide the fraction in this case 3/8 = 0.375 and then add the 115 to find you need to set your Canvas to be 115.375 inches width by 38 inches height.
To be fair the more accurate way to write it would be 115-3/8" but most people are lazy and leave off the hyphen.
Trim size = Final cut size without the bleed.
Your canvas should be 115 3/8”w x 38”h (115.375 inch x 38 inch) OR 293.05cm x 96.52cm; to this you should add at least 1/4" (0.25") of bleed on each side (or 6 mm on each side).
The total of your canvas should be minimum 115 7/8" x 38 1/2" (115.875" x 38.5") OR 294.32cm x 97.79cm (this includes a bleed).
The poster will be cut to the final size of 115 3/8”w x 38”h (293.05cm x 96.52cm)
See How can I determine how much bleed to use? for more details about bleed.
You mentioned "trim size" in your question but if you were not told specifically it was a trim size, then make sure to keep an inside safe margin of 1/2" (0.5" or 13mm) on all sides within your layout! This way even if the dimension you were given include a bleed already, there is less chances part of the design will be cut out.
Of course, if you're not certain, the best way to know is always to make sure the printer is talking about "trim" size or the layout size, and to ask the question directly or check on their website if they have one! But trim size means final size once cut.
How to convert imperial units to metric units
It's confusing to work with 3/8, 1/16 fractions and the imperial units if you're used to the metric system.
To know what's the decimal of these numbers, you can always use 1 and divide it by the biggest number on the bottom and then multiply by the small number on top.
Example: 3/8 = 1 divided by 8 then multiplied by 3 = 0.375
Once you have this, you can use Google to convert anything! You simply type your units to unit conversion or the exact numbers you want to convert.
Method 1) Unit to unit conversion in Google
Method 2) Exact conversion in Google
I don't know if you prefer centimeters or inches but this should help you find any measurement or convert anything you need.