How might you explain to a new student of English what is happening in such sentences as these?

Terry is in the library.
Terry may be in the library.
Terry might be in the library.
Terry might have been in the library.
Terry could be in the library.
Terry can be in the library.
Terry should be in the library.
Terry will be in the library.

First of all, I'd explain that in English, the name of any language (including English!) has a capital first letter.

Then, I'd go to Dave's ESL Cafe website and go through all the lessons regarding the various tenses of verbs:
http://www.eslcafe.com/grammar.html

What would YOU do?

To explain what is happening in these sentences, we need to understand the concepts of tense, modality, and certainty in English.

1. Terry is in the library.
This sentence describes Terry's current location in the library. The verb "is" in the present tense indicates that the action is happening right now.

2. Terry may be in the library.
3. Terry might be in the library.
4. Terry might have been in the library.
These sentences introduce the concept of modality, which is used to express possibility or uncertainty. The verbs "may," "might," and "might have been" indicate that there is a possibility that Terry is in the library, but it is not certain.

5. Terry could be in the library.
6. Terry can be in the library.
The verbs "could" and "can" also express possibility, but they convey a stronger sense of capability or permission. These sentences suggest that Terry has the ability or permission to be in the library.

7. Terry should be in the library.
The verb "should" expresses an expectation or advisability. This sentence implies that it is expected or recommended for Terry to be in the library.

8. Terry will be in the library.
The verb "will" indicates future tense. This sentence suggests that Terry has definite plans or intentions to be in the library at some point in the future.

When explaining these sentence structures to a new English student, you can focus on clarifying the meaning and indicating the use of different verb forms to express time, modality, and certainty.