Denpa (電波) literally means Radio waves (or any other kind of radiation coming from electronic devices). Therefore (as a slang) a "Denpa" (or Denpa-kei 電波系) person is someone who looks like he's constantly receiving and transmitting radiowaves; usually weird and delusional individuals who don't try to connect with people around them and act in erratic and incomprehensible ways. They could be seen as incoherent, creepy, or insane people.
It is accepted that the term was born in Japan when the perpetrator of the Fukugawa Serial Murder Incident in 1981 was asked for the motivations behind his acts. He claimed that he was being controlled by radio waves, leading him to commit the murders. The case got well known, and people adopted the word Denpa-kei (電波系) to describe individuals who could hear voices in their head, see things, were delusional, creepy, or insane.
These brainwashing radio waves got known as "poisonous radiowaves", or Dokudenpa (毒電波).
At the time the heavy metal band KING-SHOW started making songs referencing this incident (like Mousou no Otoko) and later the whole denpa-kei theme (like HAPPY ICE CREAM, a song about a guy who saw dead girls running around him and mocking him). These could be considered the first examples of Denpa songs.
Later the term "denpa" was also used to classify literary works like Welcome to the NHK or Shizuku, as they touched similar themes and/or had denpa-kei characters.
UNDER17 was one of the most popular bands who made musically cute songs but had lyrics referencing underground akiba and otaku culture, and very real and denpa related topics.
Thanks to them Denpa songs aren't as frowned upon as before, but since they're still too creepy for normal people, and since most otaku could be described as "denpa" people, Denpa songs are now widely associated to otaku culture, Akihabara and its technological/"electronic" image.
Denpa Song (電波ソング) is a term derived from the aforementioned Dokudenpa/Denpa-kei term, as songs that a "denpa" person would make and/or listen to. It refers specifically to songs that are either nonsensical, brainwashing, and/or touch denpa or otaku-ish themes directly or indirectly. They are often very self-referencing in nature and they intentionally blend together over-used tropes in a chaotic and exaggerated way. Parody songs are common.
Lyrically, they typically touch nerdy and otaku-ish themes, and things like delusions, telepathy, or related to madness in some way. Many have nonsensical, weird, chaotic, and/or repetitive lyrics, and many even get to the point of being creepy.
Musically, they're usually "over-the-top" and many feature things like repetitive chants, off-key singing, and other tropes. They're very addictive, and thus are seen as "poisonous".
A known example is Neko Mimi Mode as the lyrics just repeat "Neko Mimi Mode" over and over until the song ends; or Os-Uchuujin which is about a weird girl who talks about the first guy who was able to receive and understand her telepathic "radio waves". Both songs can also be extremely addictive, and most importantly, they are over-the-top, exaggerated and/or chaotic. For all these reasons, they give the overall impression of "brainwashing". That, in a nutshell, is what constitutes a denpa song.
Posted explanation in archived thread
Article about denpa aesthetic@Altair and vega
Denpa in Wikipedia
Denpa-kei in Japanese Wikipedia
Denpa song in Japanese Wikipedia
Denpa song in Nicovideo Wiki
This is a listing of acts and works related to denpa culture, and/or have denpa-kei traits, be it plot or characters. Please note that this list is not official in any way but is based solely on online comments and reviews. Please contact if you feel anything should be added or removed.
KING-SHOW / Kinniku Shoujo Tai
MOSAIC.WAV
UNDER17
KOTOKO
nomico
ave;new project
Side Protea
Toromi
Alien Nine
Arakawa Under the Bridge
Denpa Kyoushi
Denpa Onna to Seishun Otoko
Kyouran Kazoku Nikki
Sayonara Zetsubou Sensei
Welcome to the NHK!
Yuri Seijin Naoko-san
Chaos;Head
Cross†Channel
Higurashi no Naku Koro ni
Shizuku
Sayounara wo Oshiete
Subarashiki Hibi