The fundamental building block of formal language is the 'sentence'.

What is a sentence? We all use them; we probably think we know all about them.
But what is a sentence?

Most beginning students first answer this question with a statement like,

"It is a bunch of words...".

Then, after some more thought, students reach into their past knowledge and say,

"It must have a subject, it must have a verb...".

Once or twice I actually heard the answer,

"A sentence is a complete thought."

This last is the definition I would like you to retain (remember).

For example, this is a sentence:

"Students must learn all of the rules about writing sentences."

This example has all the components of a complete thought:

'Predicate' is the term used for the verb that expresses the action (feeling or thought) of the subject of the sentence.

"What" the students must learn is called the direct object; in this case the direct object is the phrase (a group of words without a verb) "all of the rules about writing sentences".

And this is a sentence:

"Students must learn all of the rules about writing sentences in order to be able to express themselves in formal English."

The words "in order to be able to express themselves in formal English" form a dependent clause that modifies (or further explains) the independent clause (sentence) "Students must learn all of the rules about writing sentences."

Please note that a phrase or even a single word could constitute a sentence fragment as well.

For example, these are not sentences:

a. "The way to school" is not a sentence because there is no verb ('predicate')!

A group of words like this is called a "phrase".
One could write, "That is the way to school." and it would be a sentence.

b. "The cottage along the shore" is not a sentence, again because there is no verb! One could write:

"The cottage along the shore was large enough for everyone." and it would be a sentence.

The fragment is a serious breach of (failure in) formal communication because it does not communicate a complete thought: the reader is not supposed to guess at the meaning of the words. This means that it the thought is vague (unclear) and leaves much room for the reader to interpret (guess) what you are really trying to say.

Fragments are, however, very useful in informal settings. When you are conversing with a friend, for example, and discussing sentences, your conversation might proceed as follows:

You might say, "I know what a sentence is." [This is a sentence.]
Your friend might ask, "What is it?" [Also a sentence.]
You might answer, "A subject and a verb." [A sentence fragment.]

Notice that the last statement is not a sentence because there is not any verb. It is a sentence fragment. Nevertheless, in the course of ordinary conversation, the statement, "A subject and a verb." is sufficient to answer the question. Formally stated, this would be:

Formal: "A sentence is a complete thought that contains a subject (noun) and a predicate (verb)."

This example serves to illustrate how, in formal communication, one cannot skip elements, leave out important words, or attempt to communicate in a short hand manner. Formal communication does not allow for short cuts.

Once again, this is necessary because formal language is used to communicate complex thoughts that require specific information with precise definitions. Informal language is sloppy because we are not trying (attempting) to be precise.

3. Run-on sentences:

A run-on sentence is one that attempts to express more than one thought at the same time.

This sentence contains several ideas, going from one thought to the next without stopping (ending a sentence). Reading it out loud almost makes one breathless. This is a run-on sentence, as it seems to 'run on' without end. The reader does not get a chance to stop and think about what is being said. To fix a run-on sentence, simply divide the different ideas into separate sentences.

"The man went to the store and bought some fruit, including apples and oranges. He then went to the next store and picked up his dry cleaning, which consisted of two suits and a dozen shirts. Following that, he had lunch with an old friend from high school. Lastly, he went home to see his wife and two children because it was his younger daughter's birthday."

4. Compound (or complex) sentences:

Please do not confuse run-on sentences with compound or complex sentences. A compound sentence uses a conjunction for example: (and, or) to combine two independent clauses (or two short sentences that can logically be combined into one).

Compound sentence:

You should begin your writing with simple sentences and then progress to writing more complex sentences. This is part of the 'learning curve', the process of acquiring new skills and knowledge that all human beings experience.

At first, it is difficult to learn the new words and rules. The learning curve is said to be 'flat'. After a while, as one becomes increasingly skillful, the learning comes faster. When this happens, and the student learns more rapidly, the learning curve is said to become 'steeper'.

For more review of these and other English grammar issues, see your grammar handbook.

In other words, Look it up!