You could also use the AlignmentMaker. They have a tutorial in which they show one example that might help you get things aligned. So, to show how to get the latex sample that you show in your post you can start by using this:

TraditionalForm[
    HoldForm[
        Row[{
            Column[{0, (t - Subscript[x, min])/(Subscript[x, max] - Subscript[x, min]), 1}], 
        ",    ",
        Column[{t <=\[AlignmentMarker] Subscript[x, min], Subscript[x, min] <= t <=\[AlignmentMarker] Subscript[x, max], t > \[AlignmentMarker]Subscript[x, max]}, Alignment -> "\[AlignmentMarker]"]
        }]
    ]
]

The alignment marker is invisible by default. You can set them to be displayed with

SetOptions[$FrontEnd, ShowInvisibleCharacters -> All]

And turn it off with

SetOptions[$FrontEnd, ShowInvisibleCharacters -> None]

The idea here is to create a grid. So we start with a row which contains 3 columns. The first two columns do not require alignment but the last one does. So we can use the shortcut EscamEsc to place them where we want them just as we would in latex. Finally, do not forget the option Alignment in the column that we want to be aligned.

Here is a screenshot of the results:

To add the piecewise bracket use what Rojo has mentioned in his answer.

You could use the \[AligmentMarker] character. To help you better write equations, lets forget about the DisplayFormula style and lets create our own.

#Set up:#

Let us create the style DisplayMath. In your notebook where you are writing your displayed equation go to Format > Edit Stylesheet.... Here you will create a new cell. Show the expression of the cell by going to Cell > Show Expression or by pressing commandshifte in a mac. Now enter the following in that expression:

Cell[StyleData["DisplayMath"],
    ConversionRules->{
        "TeX" -> {"$$\n", Convert`TeX`BoxesToTeX[#]& , "\n$$"}, 
        "HTML" -> {"$$\n", StringReplace[Convert`TeX`BoxesToTeX[#], {"<" -> " \\lt ", ">" -> " \\gt "}]& , "\n$$\n\n"}
    }, 
    DefaultFormatType->TraditionalForm, 
    TextAlignment->Center, 
    LanguageCategory->"Formula", 
    AutoSpacing->True,
    ScriptLevel->0,
    AutoMultiplicationSymbol->False,
    SingleLetterItalics->True, 
    SpanMaxSize->Infinity,
    CounterIncrements->"DisplayMath",
    FontFamily->"Times",
    FontSize->16, 
    ShowInvisibleCharacters->All, 
    UnderoverscriptBoxOptions->{LimitsPositioning->False}, 
    GridBoxOptions->{
    GridBoxAlignment->{
        "Columns" -> {{"\[AlignmentMarker]"}}, "ColumnsIndexed" -> {}, 
        "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}
    }
]

This is the style that I'm using to display mathematics. Notice the use of ShowInvisibleCharacters, this makes the \[AligmentMarker] to be displayed. If you do not wish for this to be displayed when you print or in a presentation then turn it off when you use another environment. You do so by creating a cell with these contents:

Cell[StyleData["DisplayMath", "Printout"],
    ShowInvisibleCharacters->None
]

The option GridBoxOptions simply say that the grids in this style will be aligned using the \[AlignmentMarker]. Now lets type some math.

#Writing Aligned Equations:#

Start by creating a cell using the DisplayMath style, you can do this by pressing command0 in a mac use alt in windows and entering DisplayMath. Enter a+b+c=d. What we want now is to go one step down, we can do this by pressing controlreturn. If you did this now you should see a box right below "d", we want a box to be below the whole expression. So delete that box, select the whole expression and try again. After that enter the following: c=d-a-b. This is what you should see now:

To make them become aligned, now we can use the \[AligmentMarker]. We want them to be aligned along the equal signs we can enter EscamEsc next to the equal signs. Now we should see this:

After you finish with a column, you can create another column by pressing control,. And repeat. Here is a display of your equation from your example:

To put the bracket I used what Rojo mentioned. I typed EscpwEsc at the beginning of the equation.