Q:

When to capitalize the name of a season

Which sentence uses the correct capitalization for the name of the season?

I will travel in the summer of 2013.

or

I will travel in the Summer of 2013.

A:

In contemporary English, you now normally capitalize the names of the seasons only when they are personified.  
For personification, think of “poetic” or “creative” uses, where the season is thought of as possessing the characteristics of a person, and so becomes  a proper noun meriting a capital letter.

In March, Spring shows her joyous moods.
All Nature hides from Winter’s icy breath.
We will muse on Summer’s ploys:
How no partial gifts are hers,
But now the palms and now the first
Are dozed with kisses balmy-sweet
From lips which breathe a pulsing heat.
— Charles Mair
O WILD West Wind, thou breath of Autumn's being    
Thou from whose unseen presence the leaves dead    
Are driven like ghosts from an enchanter fleeing,
— Percy Bysshe Shelley
If Winter comes, can Spring be far behind?
— Percy Bysshe Shelley
Where are the songs of Spring? Ay, where are they?  
    Think not of them, thou hast thy music too,—  
While barred clouds bloom the soft-dying day, 
    And touch the stubble plains with rosy hue; 
— John Keats

Also, Midwinter and Midsummer are often capitalized, as they are specific days.
Finally, the named seasons of a named year are also capitalized:

Back during Spring ’09, we had snow all the way till May.
We aren’t running the class for the Fall Semester.

