Shavuot, which begins this year on Saturday night, 6/11, is a major Jewish holiday in Torah, but outside of Orthodox and Hasidic Judaism, it has become somewhat less emphasized than Rosh Ha-Shanah, Yom Kippur and Pesach (and to a lesser extent, Hanukah and Purim).

“The Torah was given by G‑d to the Jewish people on Mount Sinai more than 3300 years ago. Every year on the holiday of Shavuot we renew our acceptance of G‑d’s gift, and G‑d “re-gives” the Torah.
The word Shavuot means “weeks.” It marks the completion of the seven-week counting period between Passover and Shavuot.
The giving of the Torah was a far-reaching spiritual event — one that touched the essence of the Jewish soul for all times. Our sages have compared it to a wedding between G‑d and the Jewish people. Shavuot also means “oaths,” for on this day G‑d swore eternal devotion to us, and we in turn pledged everlasting loyalty to Him.
In ancient times, two wheat loaves would be offered in Holy Temple. It was also at this time that people would begin to bring bikkurim, their first and choicest fruits, to thank G‑d for Israel’s bounty.
On this day G‑d swore eternal devotion to us, and we pledged everlasting loyalty to Him
The holiday of Shavuot is a two-day holiday, beginning at sundown of the 5th of Sivan and lasting until nightfall of the 7th of Sivan. (In Israel it is a one-day holiday, ending at nightfall of the 6th of Sivan.)” [1]

Shavuot occurs 7 weeks +1 day after Pesach.

The Christian holiday of Pentecost occurs 7 weeks + 1 day (or 50 days) after Easter.

Is there any connection?

“The Christian Pentecost…refers to the occasion of the descent of the Holy Spirit…as described in the Acts of the Apostles 2:1–31. According to Luke 22:12–13, the Descent of the Holy Spirit took place while the Apostles were celebrating the Jewish day of Shavuot (Hebrew: שבועות‎‎, lit. “Weeks”), the Feast of Weeks, a prominent feast in the calendar of ancient Israel celebrating the giving of the Law to Moses at Sinai. Subsequently, the term Pentecost may refer to the Pentecost of the New Testament and Shavuot of the Old Testament. The Shavuot of the Old Testament is a significant event shared by Jewish and Christian traditions but is not commonly celebrated as a separate holiday by Christians.
…The Pentecostal movement of Christianity emphasizes direct personal experience with God, akin to the Descent of the Holy Spirit upon the Apostles [at the first Pentecost].
…Pentecost is the old Greek and Latin name for the Jewish Festival of Weeks (Hebrew חג השבועות Chag ha-Shavuot) which can be found in the Hebrew Bible.
…Since Shavuot occurs 49 days after the first day of Passover (i.e. the 50th day, including Passover itself), Hellenistic Jews gave it the name Pentecost (πεντηκοστή, “fiftieth day”)…
…According to Jewish tradition, Pentecost commemorates God’s giving of the Ten Commandments at Mount Sinai, 49 days after the Exodus.
Orthodox churches [e.g. Greek Orthodox; Russian Orthodox, etc.] are often decorated with greenery and flowers on this feast day, and the celebration is intentionally similar to the Jewish holiday of Shavuot, which celebrates the giving of the Mosaic Law.” [2]

The “Pentecostal” movement within Christianity seeks to go back to that initial, formative spiritual experience of the early church.

Further connections:

Shavuot is one of the (3) “pilgrimage” festivals in Torah. The other two are Pesach and Sukkot. On these days, Jews were expected to travel to the Mishkan — later to the Temples in Jerusalem.

The original Christian Pentecost occurred in Jerusalem. Why were the early Christians there on that day? 

At that time, Christianity — which it was not yet called — was not looked on as a “separate religion.” They still saw themselves as a group within Judaism. Thus, they had come to Jerusalem to celebrate the pilgrimage festival of Shavuot, as mandated in Torah.

Praying in an “upper room” of a house in Jerusalem, the 120 members were filled with the “Holy Spirit.” 

This is reflected in a popular Catholic hymn for Pentecost:

“Veni Sancte Spiritus” (“Come, Holy Spirit”)

Come, Holy Spirit,
send forth the heavenly
radiance of your light.
Come, father of the poor,
come, giver of gifts,
come, light of the heart.
Greatest comforter,
sweet guest of the soul,
sweet consolation.
In labor, rest,
in heat, temperance,
in tears, solace.
O most blessed light,
fill the inmost heart
of your faithful.
Without your grace,
there is nothing in us,
nothing that is not harmful.
Cleanse that which is unclean,
water that which is dry,
heal that which is wounded.
Bend that which is inflexible,
fire that which is chilled,
correct what goes astray.
Give to your faithful,
those who trust in you,
the seven-fold gifts.
Grant the reward of virtue,
grant the deliverance of salvation,
grant eternal joy.

This hymn is about 1000 years old (with multiple musical settings, some going back to Gregorian chant). It has even been included in some Protestant hymnals, given its inspirational character.

A version in Gregorian Chant (with Gregorian notation displayed) can be seen/heard at:

https://www.youtube.com/watch?v=Z6hqAfsHURo

But the Erev Shabbat hymn, “L’cha Dodi,” written 500 years later in Sfat, Israel by the Kabbalist Rabbi Shlomo Alkabetz, also with multiple subsequent musical settings, is very reminiscent of “Veni…”, albeit with different cultural and religious idioms.

Come, My Beloved

(used as a chorus in Ashkenazic synagogues):
Come, my beloved, to meet the bride;
let us welcome the presence of the Sabbath.

“Observe” and “Remember the Sabbath day,”
the only God caused us to hear in a single utterance:
the Lord is One, and his name is One

to his renown and his glory and his praise.

Come, let us go to meet the Sabbath,
for it is a well-spring of blessing;
from the beginning, from of old it was ordained,
— last in creation, first in thought.

O sanctuary of our King, O regal city,
arise, go forth from your overthrow;
long enough have you dwelt in the valley of weeping;
truly He [God] will have compassion upon you.

Shake the dust from yourself,
arise, put on the garments of your glory, O my people!
Through the son of Jesse the Bethlehemite [the Messiah],

draw You [God] near to my soul, redeem it.

Arouse yourself, arouse yourself,
for your light is come:
arise, shine; awake, awake; give forth a song;
the glory of the Lord is revealed upon you.

Be not ashamed, neither be confounded.
Why are you cast down? Why are you disquieted?
The poor of my people trust in You [God],
and the city [Jerusalem] shall be built on her own mound**
[**the Temple Mount; i.e. the Temple will be rebuilt].

And they that spoil you shall be despoiled,
and all that would swallow you shall be far away:
your God shall rejoice over you,
as a bridegroom rejoices over his bride.

You shall spread abroad on the right hand and the left,
and you shall reverence the Lord.
Through the offspring of Perez [the Messiah]
we also shall rejoice and be glad.

Come in peace, you crown of Your Husband [God],
with rejoicing and with cheerfulness,
in the midst of the faithful of the chosen people:
Come, O bride, come, O bride.

(A very good explanation of “L’cha Dodi” can be found at:
http://www.algemeiner.com/2013/08/15/carlebach-rabbi-explains-the-meaning-of-the-lecha-dodi-prayer/)

“L’cha Dodi” mentions Shabbat as ordained in the 10 Commandments — the giving of which is commemorated on Shavuot. As Shavuot parallels Pentecost, this again suggests the parallel between the giving of the Commandments and the direct personal experience of God — re-experienced as the Shechinah (Divine Presence; the kabbalistic sefirah of “Malchut,” also indicated as the “Shabbat Queen”) on Shabbat and in the Temple, too.
In fact, in the final verse, “Come in peace” (“Bo’i v’shalom”), we bow to the West? Why? Because in the Temple, the Western end was where the Divine Presence was said to “reside” and be accessible — in the Holy of Holies, in the space between the cherubim on the cover of the ark containing the 2 stone tablets Mosheh had brought down from Mt. Sinai. Thus, bowing to the West, we are literally bowing to the “Presence” — i.e. our own direct personal experience of God.

It’s relevant in this regard to note that the HaBaD Hasidic text “Tanya” is divided into daily readings. The reading for Shavuot is the opening of the “Sha’ar ha-Yichud…” section, which declares that the diversity or duality of creation is only in human perception. From God’s viewpoint, there is — always has been; always will be — nothing other than God. Ein od.

Shavuot commemorates the giving of Torah. Pentecost commemorates the receiving of the direct personal experience of God.

They are related themes. The giving of Torah wasn’t simply about the words or mitzvot themselves. It was a “revelation” — a direct experience of God as present in Creation.

Thus, there’s a deep historical connection and thematic resonance between Shavuot and Pentecost.

On Shavuot, commemorating the display of the Divine giving of Torah, we affirm that the Divine is no less present with us now.

As Rebbe Nachman of Breslav teaches:

“The joy of the three major festivals, Pesach, Shavuot and Succot, gives us a share in the Inner Light of God, bringing new life to the mind and soul and enhancing our perception of God.” [4]

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[1] from: http://www.chabad.org/library/article_cdo/aid/609663/jewish/What-Is-Shavuot.htm

[2] from: https://en.wikipedia.org/wiki/Pentecost

[3] Rebbe Nachman of Breslav; Likutei Moharan; I:30