This passage is found in the Revised Common Lectionary for Lent, Year A and All Saint's Day, Year B (Most recently for April 2, 2017); The All Saints reading is shorter, verses John 11:32-44.
Summary: This emotional passage does not need to deep exegesis to understand. But as always, the Greek amplifies the emotions, especially of Mary. Furthermore, the Greek offers some poignant connections to other parts of John's Gospel.
ερχομαι & οραω (11:32 and 11:34, also 1:39; 1:46; 4:29; 19:33; 20:8 "Come and see"). These two verbs come together a number of times in John's Gospel. A quite impressive list actually:
A) When Jesus begins his ministry and calls his disciples.
B) When the woman at the well returns to her hometown to invite others (different cognate for "come");
C) When they bring Jesus to Lazarus' tomb.
D) When they find Jesus on the cross.
E) When they come to the empty tomb.
John's Gospel invites us to come and see, even Jesus on the cross and finally the empty tomb. The result of coming and seeing is believing.
In this passage, however, the two words come together in two very emotional ways. The more obvious one is when they invite Jesus to see the tomb of Lazarus. The more subtle one is that Mary came (ηλθεν) and saw (ιδουσα; note feminine participle endings may be more difficult to spot, sadly). In this case, she falls at Jesus feet (see next note). She has done what a disciple should do, she has come and seen. What happens when we come and see, not in intellectual or hopeful curiosity, but in grief?
ποδος (from πους, meaning "foot" as in words that have "pod" in them; 11:32) Mary will fall to Jesus feet twice in a short time. Mary cries at Jesus feet in this story; after her brother is revived, she will fall to Jesus feet to anoint them. Twice she worships at Jesus feet; the first in lament for her situation; the second in lament for Jesus situation.
Two other powerful scenes happen at Jesus feet. The other is when the women (including Mary) gather at the foot of the cross. The last is when Mary (Magdalene) stoops down to where Jesus' feet were in the tomb. Also, in chapter 13 of John's Gospel, Jesus will wash the disciples' feet. In short, if there are feet involved, it is likely an emotionally charged passage, relating to the profound cruciform servant-hood of Christ and his followers!
κλαιουσαν (from κλαιω, meaning "weep" 11:33) Simple point: People in the Bible cry. We give so little permission for people to cry today. Jesus himself cries here (11:35; it is a different word, εδακρυσεν) but don't get caught up in that. Death produces tears even from the Lord of Life.
ει...αν (if, if; 11:33) Mary has a particularly harsh construction of Greek here for Jesus. This combination of ει...αν indicates "a hypothetical that is actually false." In short, it should best be translated, "If you had been here, WHICH YOU WERE NOT, my brother would not have died."
εμβριμωμενος (from εμβριμαομαι, meaning "snort in", 11:38) This word means "admonish in anger" visually in the sense of a "horse snorting." I think its this word that has given rise to all sorts of terrible interpretations that Jesus is really mad in this passage that they don't believe. I think this is kind of nuts. I think a better translation is simply this: "Jesus was so worked up about this death that he looses control of his breathing..." To put it another way, Jesus does the uncool thing of lose control of his emotions. Unlike the rest of humanity, when Jesus' gets angry, no one is hurt, but the deepest emotion, love, and deepest power, resurrection, is unleashed.
Lastly, if this passage were not for all Saints, it might be worth focusing on what it means to unbind Lazarus, but I think the interaction of Mary is where its at!