I get 'Failed to execute 'detect' when running my Minecraft command?

Basic troubleshooting

First of all, let's talk about your command here

/execute @a ~ ~ ~ detect ~ ~ ~ minecraft:stone_button 0 /scoreboard players >set @a slow 2

Everything up to the 0 is fine, but not very useful (see below). The command fails because the syntax for the scoreboard part is wrong. First of all, the > is a syntax error.

Secondly, make sure that the scoreboard objective called slow actually exists by executing

/scoreboard objectives add slow dummy

What it does might not be what you want

Now, regarding what this command does when it works, and why else it might fail:

It will look for a stone button with data value 0, i.e. one facing downwards, at the position of the feet of every player on the server. While the command itself would work, detecting this is highly unlikely since it involves someones head being inside a block (the bottom of which the button is one), or a floating button. What you are likely looking for data value -1, which allows all orientations of the button (If not, check the linked wiki page for the correct orientation, or change the coordinates at which you are trying to detect.)

Furthermore, if you use @a for the scoreboard part, you will set everyone's slow score to 2, not just that of the person at the button. Use @p to only affect the player closest to the coordinates specified in execute, which is exactly at the position of the player. If you want to change the position where the button has to be (i.e. to the head of the player), change the second set of coordinates.

Fixed command, looking for any stone button at any player's heads:

/execute @a ~ ~ ~ detect ~ ~1 ~ minecraft:stone_button -1 scoreboard players set @p slow 2