What are members in dimension hierarchies?

All of those are members.  Members are the named elements in the dimension.  So 2009, 2009-January, 2009-January-01 are all members.  You can define them moe specifically by their position in the hierarchy.  For instance 2009-January-01 is a level 0 member or leaf member.  2009-January would be considered a level 1 member, 2009 would be considered a level 2 member.

You can imagine a simple cube with two "dimensions" as a grid in excel with dimension A along one side, and dimension B along the other.  Each "column" or "row" of the grid corresponds to a "Member".  The cells in the middle of the grid (where the numbers are) are referred to as "Tuples", with each "tuple" being referencable by the members along each edge.  

Levels in a dimension are (usually) simply a 'calculated' member that represents the total of its "children".  So in your example, "January 1 2009" and "January 2 2009" are both members at the "Day" level of your cube.  "January 2009" is another member, but its at the "month" level, and simply represents the total of all the "day" members below it.  "2009" is likely also a member, and there will usually be another member above that which represents "All" dates in the cube.  

To go back to my spreadsheet example, imagine your cube has all the dates down one side.  If you were to insert a row after "January 31, 2009", and then add in a "Sum()" function to add up all the values from "January 1, 2009" to "January 31, 2009", you'd have a new row (member).  Effecitvely, you'd have created a "month" level in your spreadsheet.

Hello:

After reading the comments and the SSAS tutorial my doubts remain. I will try to be more precise:

This is my dim table (slightly modified):

ID    Year    Month        Day
----   -----     -----          ------
1    2009    January     01
2    2009    January     02
3    2009    January     03
4    2009    February    01
5    2009    February    02
6    2010    January      01
7    2010    January      02
8    2010    January      03
9    2010    February     01
10   2010    February     02

1) The table has four columns, the ID one being the Primary Key.

2) Every column corresponds to an only attribute in the dimension (so not having composite keys).

Then, some questions:

a) Would you say that only "1", "2", etc. (the items in the ID column) can be called members while the other items in the table, such as "January" or "2009" can't be?

b) Would you say that a member is really  the whole set of items in a row, for instance "1    2009    January     01"?

c) Is "02" in the Day column/attribute a member of the dimension? Note that I refer to "02", not to "02-Januay-2010" or "02-January" or similar.

   c.1) Does besides exist a member such as "02-January" or "02-January-2010"?

Thanks,

Juarrero