Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$:

$xP(x-c) = (x - 2014)P(x)$
1. **Initial Setup and Contradiction for \( c = 0 \)**:
   - Given the polynomial \( P(x) \) with real coefficients such that \( P(2014) = 1 \) and \( xP(x-c) = (x - 2014)P(x) \) for some integer \( c \).
   - Assume \( c = 0 \). Then the equation becomes:
     \[
     xP(x) = (x - 2014)P(x)
     \]
   - This implies:
     \[
     xP(x) = xP(x) - 2014P(x) \implies 2014P(x) = 0
     \]
   - Since \( P(2014) = 1 \), this is a contradiction. Therefore, \( c \neq 0 \).

2. **Proving \( c \) is a Divisor of 2014**:
   - Suppose \( c \) is not a divisor of 2014.
   - For \( x = 2014 \), we get:
     \[
     2014P(2014 - c) = 0 \implies P(2014 - c) = 0
     \]
   - For \( x = 2014 - c \), we get:
     \[
     (2014 - c)P(2014 - 2c) = 0 \implies P(2014 - 2c) = 0
     \]
   - By induction, for \( x = 2014 - nc \), we get:
     \[
     P(2014 - (n+1)c) = 0
     \]
   - This implies \( P \) has infinitely many roots, which means \( P(x) \equiv 0 \), a contradiction since \( P(2014) = 1 \).
   - Therefore, \( c \) must be a divisor of 2014.

3. **Form of the Polynomial**:
   - Let \( 2014 = ac \) where \( a \) and \( c \) are integers.
   - For \( x = 2014 + kc = c(a + k) \), we get:
     \[
     P(c(a + k)) = \binom{a + k}{a}
     \]
   - This implies:
     \[
     P(x) = \binom{\frac{x}{c}}{a}
     \]
   - Since \( P(2014) = 1 \), we have:
     \[
     P(2014) = \binom{a}{a} = 1
     \]

4. **Verification**:
   - Verify that \( P(x) = \binom{\frac{x}{c}}{a} \) satisfies the given functional equation:
     \[
     xP(x-c) = (x - 2014)P(x)
     \]
   - Substitute \( P(x) = \binom{\frac{x}{c}}{a} \):
     \[
     x \binom{\frac{x-c}{c}}{a} = (x - 2014) \binom{\frac{x}{c}}{a}
     \]
   - Simplify using properties of binomial coefficients:
     \[
     x \binom{\frac{x}{c} - 1}{a} = (x - 2014) \binom{\frac{x}{c}}{a}
     \]
   - This holds true, confirming the form of \( P(x) \).

5. **Conclusion**:
   - The polynomial \( P(x) \) is given by:
     \[
     P(x) = \binom{\frac{x}{c}}{a}
     \]
   - Where \( c \) is a positive divisor of 2014.

The final answer is \( \boxed{ P(x) = \binom{\frac{x}{c}}{a} } \) where \( c \) is a positive divisor of 2014.