Eridan has some nifty white magic powers, which he believes to be purely scientific in origin.
The powers aren't reliant upon the wand he wields, but he believes they are, so without it he just...can't do shit.
Although not as versatile as the powers of some of the other homestuck characters, they are still OP as fuck.
With them, he can do the following.

In general, he can shoot white magic offensively, as either a large wall of energy or a bolt.

The bolts are also INCREDIBLY fast, as they managed to pierce Kanaya(below) near-instantaneously.

He's also displayed the ability to levitate using his powers, by surrounding himself in an aura of white magic and floating into the air.
The aura can probably act as somewhat of a shield of sorts, burning up things that come into contact with it.
Now I don't mean it's some kind of all-purpose 360 degrees impenetrable shield of magic...just offers some protection. Some.

The main thing to keep in mind about his powers is just how potent the blasts are.
To put it into perspective, first we can take a look at Eridan's original weapon, Ahab's crosshairs.

It's the huge blue rifle he's carrying, not the seahorse, numbnuts.
In any case this is a good example of how powerful it is.





Later on, Eridan attempts to duel Sollux with the crosshairs going up against Sollux's telekinetic blasts.
The crosshairs is overpowered by it, and Eridan is defeated.
For a good idea of what Sollux can do -


This is Aradia sitting on the roof of Sollux's communal hiveblock.
She would be the size of your average 13 year old child.

This is Sollux's lusus, being levitated in midair above said communal hiveblock.

The white floating object is Sollux's lusus from the frame above.
That tiiiiiiiny black dot below it is Aradia, still sitting on the roof.

From a later time, Sollux's lusus sitting on the roof as meteors come a' crashin' down.

From inside the building, annnnnnnnd...

Bam.

Now, fast forward into the future when Eridan obtains the wand.
He duels Sollux again, this time using his car-aaaaayzee hope powers.
And..................




Yeah.
That pretty much covers it
and this is all there really is to say on the matter.