Both lemon juice and vinegar contain acids -- citric acid in the case of lemon juice and acetic acid in the case of vinegar. Both of these are relatively weak acids, but citric acid is slightly stronger than acetic acid. However, when people refer to a strong acid in everyday conversation, they may mean something slightly different.
Strength of an Acid
The strength of an acid is a measure of its tendency to donate a hydrogen ion when in solution. For example, when acetic acid is dissolved in water, it separates into hydrogen ions and a conjugate base called acetate. The hydrogen ions, represented as H+, bond with water molecules to form hydronium. The level of equilibrium -- that is, the point at which the acid stops giving up hydrogen ions -- determines the strength of the acid. The lower the level of acid present in a solution when this happens, the stronger the acid is. The strength of an acid is represented by the value pKa. The lower an acid's pKa, the stronger it is. (Ref 1, 2)
Although pKa measures the acidity of a substance, when most people talk about "acidity," they're referring to pH, which measures the presence of an acid in a solution. Lemon juice and vinegar are both solutions of an acid in water. Whereas pKa measures the strength of the acid itself, pH includes not only this but the amount of acid in the solution. As a result, the pH of vinegar and lemon juice doesn't correspond directly to the strength of the acetic and citric acid because the amount of acid in the solution varies. For example, two different types of vinegar will both contain acetic acid, but they may contain different amounts. Although the acid in both will have the same pKa, or strength, the two types of vinegar will differ in their pH. As with pKa, the lower the pH of a solution, the more acidic it is. (Ref 3, 4)
Lemon juice contains high levels of citric acid, with around .05 grams per milliliter. (Ref 5) Citric acid is a weak acid, with a pKa of 2.79. (Ref 6) As with any product, the exact level of citric acid in lemon juice can fluctuate. In general, however, lemon juice is highly acidic, with a pH of 2, because of the high percentage of acid it contains. (Ref 7) In some cases, lemon juice can have a slightly higher pH of up to 2.6. (Ref 9) One way to demonstrate the low pH -- and therefore high acidity -- of lemon juice is with a litmus strip or indicator solution. These substances undergo a chemical reaction which causes them to change color in contact with acids. Lemon juice dropped into an indicator solution or onto a litmus strip will cause it to turn a bright red, indicating its high acid content.
Just as lemon juice is a solution of citric acid and water, vinegar is a solution of acetic acid and water. Acetic acid is a much weaker acid than citric acid, with a pKa of 4.74. (Ref 8) The pH of vinegar can vary greatly depending on the specific type or composition. Some vinegar can be nearly as acidic as lemon juice, with a pH of around 2.4, while other types are much more basic, with a pH of over 3. (Ref 9) It is important to remember that small distances on the pH scale represent large differences in acidity. A solution with a pH of 2 is ten times as acidic as one with a pH of 3. Like lemon juice, vinegar will turn a universal indicator red, but the color will usually be a somewhat lighter orange-red, indicating its slightly higher pH. (Ref 10)