Minecraft – ‘/execute if entity’ but with relative coordinates

In the future, I recommend checking the wiki before posting a question here.

This is entirely possible using the execute command and target selector variables and arguments.

Executing commands at the player

The execute command executes commands on an entity. The @p selector selects a single player.
The name= argument only selects entities with a specific name.
So execute @p[name=<your username here>] ~ ~ ~ <command> will run <command> at your location.

testfor the nearest entity.

The @e selector finds entities of any type.
The c= argument selects a certain count. If this number is positive, it will select these in order from closest to furthest. c=1 will select the nearest entity.
The rm= argument selects only entities outside of a radius. If this is set to 0, it will select any entity that is not the executing entity.
So testfor @e[c=1,rm=0] would find the nearest entity to the executor of the command.

Putting it together

So, the first command will execute the second command at the player, meaning your finished command is

/execute @p[name=] ~ ~ ~ testfor @e[c=1,rm=0]

Closing comments/notes

As far as using testfor goes, I'd recommend against it. The scoreboard is almost always more useful, since you can use things like tag to mark the entity you found.

If you place a comparator in front of a command block with

testforblock x y z

Then it should yield a redstone signal. You may want to place a repeater in front of the comparator.

A visual guide: