How do I count the number of columns in each table?

You can look at columns in sys.columns:

Returns a row for each column of an object that has columns, such as views or tables. The following is a list of object types that have columns:

• Table-valued assembly functions (FT)
• Inline table-valued SQL functions (IF)
• Internal tables (IT)
• System tables (S)
• Table-valued SQL functions (TF)
• User tables (U)
• Views (V)

For columns count in tables, this query can be used:

SELECT [Schema] = s.name
    , [Table] = t.name
    , number = COUNT(*)
FROM sys.columns c
INNER JOIN sys.tables t ON c.object_id = t.object_id
INNER JOIN sys.schemas s ON t.schema_id = s.schema_id
GROUP BY t.name, s.name

• sys.tables can be replaced by sys.views for count in views

• sys.objects can also be used with a WHERE clause on required type(s):

  SELECT [Schema] = s.name
      , [Table] = o.name
      , number = COUNT(*)
      , o.type_desc
  FROM sys.columns c
  INNER JOIN sys.objects o ON c.object_id = o.object_id
  INNER JOIN sys.schemas s ON o.schema_id = s.schema_id
  WHERE type IN (
      'U'     --> U = Table (user-defined)
      , 'V'   --> V = View
      , ...) 
  GROUP BY o.name, s.name, o.type_desc;
  

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This would work too although it is preferable (read The case against INFORMATION_SCHEMA views from Aaron Bertrand) to use the first query:

SELECT TABLE_SCHEMA
    , TABLE_NAME
    , number = COUNT(*) 
FROM INFORMATION_SCHEMA.COLUMNS
GROUP BY TABLE_SCHEMA, TABLE_NAME;