Type of tool used to fasten wood to brick
The picture that you show is indicating some type of fastener to hold the wood to the brick. That is not really a tool. A tool may very well have been used to install the fastener in the first place. 

It is difficult to discern the type of fastener that was used from your picture. A close up of the fastener would help but from zooming into your picture it appears that the wood may have been mounted using concrete nails that were aimed into the mortar lines between the bricks. Very hard steel nails of this type are often installed using a single shot tool called a "concrete nail gun" that shoots out the nail with very high force and velocity so it lodges into the concrete with minimal chipping. These nail guns use a small caliber blanks that contain gun powder that fires off to embed the nail.  

 

Removing concrete nails can lead to quite a bit of damage to the medium that they are set into. In your case it may be necessary to cut the wood pieces away using a reciprocating type saw right along the nail. Once the wood is removed the nail may be either cut with a metal cutting blade or bent back and forth till it breaks off. 

For your new wood installation you could choose to use a similar type of concrete nail installation but I am not experienced with how they work if you try to nail right into a brick itself. The brick may very well shatter or break into pieces. There are alternative types of expansion type fasteners that would be more appropriate for mounting wood to brick. These mostly require drilling a hole for the part of the fastener that gets installed onto the brick.