Grammar

Chinese does not have a lot of grammar in the sense that we think of grammar. For example, Chinese has no tense, no masculine/feminine, no singular/plural, and verbs do not change form. Instead, it has lots of words that allow us to indicate these things. If I say "Yesterday I eat with my friend," it is clear because of the word 'yesterday,' that this took place in the past, even though the verb 'eat' has not changed its form. In the same way, if I say, "Please give me two apple," you know that I want more than one, even though the word 'apple' is still singular.

Basic word order in Chinese is the same as in English: Subject Verb Object, as in 我爱你。Wǒ ài nǐ. I love you. Interestingly, the object can be placed at the beginning of the sentence if you want to emphasize it, and often is.

Adjectives - good, bad, tall, busy, tired - can act as verbs, and are referred to as stative verbs (because they describe the state of something).
他很高。Tā hěn gāo. He (is) very tall. Note that Chinese are unlikely to say 他高。Tā gāo. He (is) tall, because they do not like one character words; they don't sound good.

Particles are placed after verbs to indicate a change in aspect, such as ongoing, completed (either earlier or in the future), etc.
Particles are also used at the ends of sentences to change meaning, such as the question particle 吗 ma and the suggestion particle 吧 ba. (Note that particles have the neutral tone, thus no tone mark on the pinyin.)

Chinese makes extensive use of measure words, also known as classifiers. Almost every noun requires a measure word if it is quantified - 4 plates; 10 dresses - or specified - this purse; that book. Note that English uses these as well: a herd of cattle; a flock of geese; a slice of bread; however we do not say 'a piece of pencil.' In Chinese, every noun requires one1. Chinese also uses measure words for quantities such as a cup (of tea), a bottle (of soda), a bag (of oranges).
Chinese makes extensive use of fixed sentence structures, sets of terms used in specific places in a sentence.[1] 

[1]There are a few nouns in Chinese that do not take a measure word, such as 天 tiān day and 年 nián year, and in some cases measure words appear to function as nouns.