So on my timetable it says...

LABR ST - 2D - Sectioning conflict tutorial
LABR ST - 2D - Sectioning conflict core
MATH - 1D - Sectioning conflict tutorial
MATH - 1D - Sectioning conflict core

I dont really know what this means or how to fix it. I know I have to do something with SOLAR when it reopens tomorrow. However, I work all day tomorrow and will have trouble dropping and adding new classes. What are the chances i will be able to get to into a class late tomorrow?

Sectioning conflict means that you have the tutorial/core conflicting with something else at the same time. So, you'll have to move the tuts/core to a different time.
This might help:
https://adweb.cis.mcmaster.ca /mtt/

Spots are usually available early in the morning/afternoon. If you can't get in today. Keep looking for openings till the beginning of Sept - you'll find random openings on days when Profs. open up a few seats / students drop courses / change stuff around.

use the link provided http://asylumartists.com/mcmaster/generator.html ( time table generator ) to prepare a conflict-free timetable before SOLAR reopens on August 9th. Take a screenshot of your cores/tut/lab sections once you've prepared your conflict-free timetable. Trust me, this will really help you organise your work before logging in to SOLAR for switching your cores/tuts around as you will have to in order to get rid of "sectioning conflict".

As Ish mentioned, if your unable to register for a specific time slot, dont panic, you've got time until September 1st. Spots will open up.