Who are humans' closest relatives, after primates?
Short answer

It is a flying lemur (there exist only 2 species). Flying lemurs and primates are together a sister clade to treeshrews.

Easy source of information

Have a look at the post The best free and most up to date phylogenetic tree on the internet?.

In short, you can have a look at onezoom.org, tolweb.org or opentreeoflife.org by yourself!

Tree of life of placental mammals

Here is a screenshot coming from tolweb.org.



In this tree, there are polytomies showing everything that is unknown. This tree was last updated in 1995 and there are also clades that we now know were misplaced including the bats (as commented by @NoahSnyder). It is not handy to make a good screenshot from onezoom.org or opentreeoflife.org so I just welcome you to explore the tree of life by yourself on onezoom.org or opentreeoflife.org to get a better representation of the reality.

Tree of Supraprimates

Supraprimates (=Euarchontoglires) include all primates and a few of the most related clades. Stealing the tree from @Gwenn's answer, here is the tree of Supraprimates

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Here is the original tree from Janecka et al. (2007)



Here we see that flying lemurs (=Dermoptera) is a sister clade of primates (ignoring the extinct plesiadapiformes) and treeshrews (=Scandentia) are a sister clade of flying lemurs and primates together. Below I briefly talk about these three clades and about their position on the phylogenetic tree.

Flying lemurs

Flying lemurs is a small clade that contains only two extant species, both found in the family of colugos. Note that flying lemurs are not lemurs (which is confusing). Here is what a flying lemur look like



The position of flying lemurs on the phylogenetic tree has always been quite uncertain. In the tolweb.org screenshot above, flying lemurs (=Dermoptera) is a sister clade to bats (=Chiroptera) and together they are sister clade to treeshrews and primates. However, Janecka et al. (2007) placed flying lemurs as the sister clade to primates. Together flying lemurs and primates are a sister clade to tree shrews.

Tree shrews

Treeshrews is a clade of 20 species native from southeast Asia. Before, the era of phylogenetic comparisons, treeshrews were mistakenly thought to be part of the Insectivora.



Extinct clade of Plesiadapiforms

Although they are represented above as a sister clade of primates, it is possible that they the fossil we found are just individuals coming from the lineage giving rise to primates. As such, they would not be a clade aside from primates. Below is an artistic representation of what they may have looked like.



Introduction on phylogenetic trees

As the question is quite introductory, I presume you might want to improve your understanding of phylogenetic trees. You might want to make sure, you understand what a monophyletic group (=clade) is and understand why a chimpanzee and a human are both equally related to any species of treeshrews.

Here are two sources of information for you:


Understanding Evolution (UC Berkeley) > The history of life
This answer on Biology.SE