In literature, an allusion is a reference to something outside of the literature which is generally considered to be familiar or recognizable, often from history, mythology, the Bible, or other literature. For example, if I were to describe a scenario in which one person was David and the other Goliath, I would be making the point that the smaller, weaker, or more unlikely person is going to bravely defeat the bigger, stronger, and perhaps more expected person. For those who know the story of David and Goliath, this allusion is a quick and effective way to paint a picture without using a lot of words. The same is true if a person says she feels as if she is wearing a scarlet letter; this is a reference, of course, to Hester Prynne in The Scarlet Letter by Nathaniel Hawthorne and suggests a feeling of public shame and/or condemnation.
For those who do not know the original references, allusions are either distracting or something they just skip over; for those who do know the references, the story (whether it is found in a television show, music, or movie) becomes richer because they are able to add their own connections and connotations to these works.
There is really no such thing as a "modern allusion" in the Bible, so I assume you are being asked to find a modern allusion--something modern that alludes (refers) to the Bible. In other words, find a movie, a song, or something else which contains an allusion (a reference to) the Bible.
One of the most well know Biblical allusions is in a song that has been popular several times and in several versions (including the movie Forrest Gump) in the past fifty years or so; "Turn, Turn, Turn" is actually a musical version of Ecclesiastes 3:1-8.
To Everything (Turn, Turn, Turn)
There is a season (Turn, Turn, Turn)
And a time to every purpose, under Heaven
A time to be born, a time to die
A time to plant, a time to reap
A time to kill, a time to heal
A time to laugh, a time to weep
A time to build up,a time to break down
A time to dance, a time to mourn
A time to cast away stones, a time to gather stones together
A time of love, a time of hate
A time of war, a time of peace
A time you may embrace, a time to refrain from embracing
A time to gain, a time to lose
A time to rend, a time to sew
A time for love, a time for hate
A time for peace, I swear it's not too late
References to Samson and Delilah can often be found in lyrics, including a song by the Grateful Dead; and of course anything that refers to a literal Heaven or a literal Hell is alluding to the Bible. Any familiar characters can be found in places other than the Bible. I have attached a link to an excellent list of common characters from literature, mythology, and history as well as the Bible. I have also attached one list of possibilities specific to the Bible, but there are certainly plenty of others from which to choose if you cannot think of any others on your own.
I'm confident you will think of at least one one yourself now that you understand what a modern biblical allusion is, as they are quite common.