Here's the short answer: yes.

Here's the grisly details:

Trigger is a static class (a class that exists in a global context, without your code defining it). It tells your code information about the state of the trigger (before insert, after update, etc), as well as the records that are affected. There are four static variables for the records being processed: old, new, oldMap, and newMap. State information are boolean functions such as Trigger.isBefore() and Trigger.isDelete().

Trigger.old and Trigger.new are List<Sobject> (a regular array with the data type of the trigger's records); Old is the value before the user initiated the transaction, new is the value that the user submitted as the new value. Old won't exist in insert and undelete triggers, and New won't exist in delete triggers.

Trigger.oldMap and Trigger.newMap are Map<Id,Sobject>, which is a "dictionary" class. Much like a physical dictionary, it has a set of "words" that it knows are associated with certain values ("definitions"). In this case, the "word", called a "key", is the ID of the record being processed, and its "definition", called the "value", is the record passed to the trigger's code.

You could consider a Map to be an array that accepts strings instead of numeric indices (a map can accept Integer as a key type, but this statement was only for metaphorical purposes). To get the keys, use "keyset()", for the values, use "values()", to see if a key is already in the map, use "containsKey(keyValue)", to get the value of a key, use "get(keyValue)", and to set a new value in its place, use "put(keyValue,dataValue)". Note that "put" won't work on the Trigger maps, because they are special. I included this for completion.

The "keys" of the maps are actually a Set<ID>. A Set is like the mathematical concept of a set (the union of numbers that are unique). For example, { 1 5 4 } is a set, while { 1 4 4 } is not a set (4 is repeated). It is similar to an array, but it knows what values it contains, and silently ignores duplicates. Sets contain methods such as "contains", to find if a certain value is in the set. From the set { 1 5 4 }, contains(5) would be true, while contains(2) would be false.

In a "before" trigger, you can use "addError" on any value to stop the record from saving, or you can modify the value in "new" or "newMap" to cause a different value to be saved to the database. You should not try to use remove, removeAll, add, addAll, etc methods that are normally available to Lists and Maps, because it won't work. In an "after" trigger, you shouldn't attempt to modify any values at all in the four static variables.

As you may have gathered, you can use triggers to do "if previous value is different than new value" checks, and make specific changes to data based on that input, and even more complex logic such as checking values against a different object, etc. If you modify the values in "new", and later throw an error, the user won't see the changes to the values (you can't use this as a "suggested correction" mechanism).

So, to completely answer the question, "Trigger" is a global class with static members, newMap is a Map<Id,Sobject> that gives you dictionary-style access to records, with a "key" for the record being processed, and its data value being the actual record, and keySet() is a function on that Map that returns a Set<Id>, which happens to be all of the ID values for all records involved in the trigger.

Suggested Reading: Apex Code Developer's Guide, in particular Set, Map, and Trigger.