As was mentioned before, Chinese is not a monolithic homogenous structure, but rather a motley concoction of at least 15 major dialects united by one recently adopted spoken standard and one written language (Traditional and Simplified scripts are only variations). There are quite a few terms both in English and Chinese used for Standard Modern Chinese, both on this site and other language-related sources. For the sake of terminological clarity, let us review the most common of them.
Putonghua -- "Normal language".
More than half a century ago the conclusion of a long-standing debate on the standard national language in China was formulated as: "Consisting of the sounds of the Beijing dialect, based on the lexicon of the Northern Chinese dialect, and the grammar of the written language (白話 bai2 hua4) used for communication between the Han people (漢民族 Han4 min2 zu2)." In English putonghua is known as Mandarin -- the word is derived from the name of Chinese emperial government officials who were supposed to use the standard language.
Beijing hua-- "Beijing speak" or "Beijing dialect"
First of all, the Beijing dialect is a dialect, though the one whereon putonghua is based. It is the way common Beijingers speak in their daily life -- with all the features typical for vernacular. Its main distinctions are:
tonal and sound variations, e.g., ”言語一聲兒“ is pronounced (yuan2 yi yi4 shengr1) instead of (yan2 yu3 yi2 shengr1)
dialectal/slang expressions like:
打住 (da3 zhu4) -- meaning 別再說了 (bie2 zai4 shuo1 le)
棒棰 (du3 chui2) -- meaning 外行 (wai4 hang2)
別價 (bie2 jia4) -- meaning 別這樣 (bie2 zher4)
二百五 (er2 bai3 wu3) -- meaning 笨蛋 (ben4 dan4)
敢情 (gan3 qing2) -- meaning 當然 (dang1 ran2) or 原來 (yuan2 lai2)
罷了 (ba4 le) -- in the sense of "Oh well!"
蓋了帽兒了 (gai4 le mar4 le) -- meaning 非常好 (fei1 chang2 hao3)
and many others
Guoyu-- National language
The term is used for the variety of Standard Chinese as accepted in Taiwan. For the most of Taiwanese, it is the medium of education rather than their native language. However, through the decades of standardized compulsory education, it has become the core of vernacular speech for the post-war generations. The younger the person is, the more likely their Standard Chinese will be fluent.
Guoyu's standard form is almost identical to Putonghua, but such typical Beijing phenomena as qing-sheng-fication (輕聲化 -- light-tone-ization) and er-fication (兒化) are much less common. Also, some consonants (most notably zh-, ch-, sh-) may be pronounced softer than in Mainland China, while the 3rd tone pronounced in Taiwan as low flat rather than the Mainland way -- sharply falling and then rising back up.
Huayu
This is the Standard Chinese -- as opposed to Chinese dialects -- used as (once again) the medium of education by ethnic Chinese in Singapore, Malaysia and other countries. It is more of a literary tradition, since the vernacular for the most of the Overseas Chinese community are various Chinese dialects. Its written form is also known as Bei-hua 白話 (bai2 hua4). Save Singapore, it almost invariably uses the Traditional script for writing.
TV Standard
In a number of countries, the TV speak is considered to be the standard language. However, Beijing TV anchorpersons are (in)famous for their unnatural forced intonations (claimed to add more pathos and dramatism to their commentaries) that they themselves never use in normal conversation.