Notation, Symbolize and InfixNotation
Notation
Notation[boxes expr]  set boxes to be parsed to expr and expr to be formatted as boxes 
Notation[boxes expr]  set boxes to be parsed to expr 
Notation[boxes expr]  set expr to be formatted as boxes 
Notation[notation, opts]  define a notation notation with the options opts 
RemoveNotation[notation]  remove the notation notation 
Syntax of notation declarations.
Notation takes both an external representation and an internal representation as arguments.
Mathematica translates any input matching the external representation into the corresponding internal representation and, reciprocally, formats any expression matching the internal representation into the corresponding external representation. In this context, representation means a composite structure made up of boxes representing some "notation".
The following declares a new notation for gplus. 
Any gplus expression is now formatted in the new notation.
Out[4]=  

Notations defined using
in their definition both parse and format expressions according to the given notation. However, you can restrict the notation to only parsing or only formatting by using
or
respectively, instead of
in your notation statements.
This defines a notation for the parsing of a hypothetical DomainIntegral. 
DomainIntegrals are now parsable.
Out[6]=  

The following defines an output format for Derivative objects that looks more like that of traditional mathematics. 
Derivatives are now formatted according to the new derivative notation.
Out[9]=  

To allow the previous output to be used as input you can define an interpretation of partial derivatives. 
You can now use these new notations for derivatives.
Out[12]=  

You should define your notations in such a way that they both parse and format, since users will generally expect this functionality.
The following notation both formats and parses arrows overscripted by Apply. 
Simple rules like linearity can now be entered in a visually intuitive way.
Out[15]=  

You may at first feel that having underscores on both sides of a
Notation statement is somewhat unsettling. However, notational transformations usually work in both directions, therefore having underscores on both sides of a notation statement is natural, and soon becomes intuitive.
Symbolize
Symbolize[boxes]  treat boxes as a symbol 
Symbolize[boxes, opts]  treat boxes as a symbol with the options opts 
RemoveSymbolize[boxes]  remove the treatment of boxes as a symbol 
Syntax of symbolization declarations.
Symbolize is used to treat a class of composite boxes internally as symbols. It is sometimes necessary and desirable to be able to symbolize composite objects; for example, to be able to attach rules to a composite object, it must be treated as a symbol.
Symbolize also symbolizes classes of expressions, for example, to symbolize all objects subscripted by Rule. 
You can use symbolized box structures as pattern variables in definitions. This lets you create formulas in a more familiar notation. However, you must use pattern expressions of the form
CompositePatternVariable :
PatternContent otherwise
Mathematica may interpret the pattern expression as
CompositePatternVariable PatternContent.
This defines the transmission coefficient for an incident wave packet on a potential step under certain conditions. 
InfixNotation
InfixNotation[op, func]  treat op as an infix operator representing the function func 
InfixNotation[op, func, opts]  treat op as an infix operator representing the function func with the options opts 
RemoveInfixNotation[op, func]  remove the composite infix operator op 
Syntax of infix notation declarations.
InfixNotation is used to treat a composite box structure as an infix operator.
InfixNotation requires both a composite box object which will represent the infix operator and a symbol which will be the full form head of the expression. A simple parallel of this duality in
Mathematica is that the infix notation
+ has the full form head
Plus.
This declares that the composite object +_{{}} should act as the infix form of Join.
Out[24]=  

The infix notation +_{{}} both parses input and formats output.
Out[25]=  

An advantage of using
InfixNotation over that of using
Notation to define an infix operator is that
InfixNotation parses an expression into a flat internal form without evaluation. Although this is a subtle difference, it is an important one (see
ring operations and
parsing without evaluation.)
The function
InfixNotation can be compared to using the infix form of functions ~
~ (cf.
Special Ways to Input Expressions)
Entering Notations
When defining your own notations it is
critically important to use the
notation templates. The notation templates are contained in the notation palette shown below.
The reason for this restriction is that the templates pasted into a notebook contain essential tag boxes embedded in the correct way. These embedded tag boxes allow
Notation ,
Symbolize and
InfixNotation to properly group and parse the new notation being defined. Further, they allow these functions to get the styling information of the new notation so that the output is formatted with the same spacing, sizes, adjustments, etc. as the input.
A blank notation template looks like the following. 
Advanced note: You can circumvent using the notation palette by wrapping the corresponding literal box structures in a ParsedBoxWrapper.
In
Mathematica it is possible to create input aliases for any notation or expression you want. The normal interface to these input aliases is rather cumbersome and so the notation package contains the function
AddInputAlias to facilitate adding input aliases to the current notebook. Each of the common templates has an alias associated with it for easy entry. For instance typing
notation in an input cell will enter the notation template
Notation[ ].
Add an input alias for apply statements to the current notebook. 
Now that we have created this alias we can enter "apply" templates by simply typing
apply.