In ancient times a seal or signet was an instrument, often metal or stone, with an engraved pattern or design on it. It would be pressed into a softer material, usually wax or clay leaving an imprint like a stamp.
A seal affixed to a document (usually on a scroll) would have to be broken in order to unroll the document and read it. This meant it was in one way like today's "tamper-resistent seals" on packages. But it also was used to verify the authenticity of who sent a letter, especially a king's decree, because it would have the King's special stamp on it, similar to what we might use a signature for today. Handing over a signet ring would be symbolic way of allowing someone else to act on your behalf with your authority. However, sealing has both literal and figurative uses in the bible. Things besides just documents could be sealed. The door to the lion's den was "sealed" to prevent unauthorized entrance after it was shut. Jeremiah sealed the deeds of the field which he bought from Hanamel as a symbol of formal radification of his transaction. In the more figurative sense, it could also be used "for the act or token of authentication, confirmation, proof, security or possession." (ISBE)
Thayer's lexicon enumerates the various uses of ἐσφραγίσθητε to include:
(1) to set a seal upon, mark with a seal, to seal
(1a) for security: from Satan
(1b) since things sealed up are concealed (as the contents of a letter), to hide, keep in silence, keep secret
(1c) in order to mark a person or a thing
(1c1) to set a mark upon by the impress of a seal or a stamp
(1c2) angels are said to be sealed by God
(1d) in order to prove, confirm, or attest a thing
(1d1) to confirm authenticate, place beyond doubt
(1d1a) of a written document
(1d1b) to prove one’s testimony to a person that he is what he professes to be
But when it comes specifically to this verse in Ephesians, the The International Standard Bible Encyclopaedia suggests the intended meaning is a mark of ownership by God:
God by His Spirit indicates who are His, as the owner sets his seal on his property; and just as documents are sealed up until the proper time for opening them, so Christians are sealed up by the Holy Spirit “unto the day of redemption” (Eph. 1:13; Eph. 4:30; 2Co. 1:22).
Or to quote Albert Barnes:
"Christians are said to be sealed; to be sealed by the Holy Spirit Eph. 1:13; Eph. 4:30; that is, the Holy Spirit is given to them to confirm them as belonging to God...God grants to them His Holy Spirit as the certain pledge that they are His, and shall be approved and saved in the last day."
Also note that in this verse, to seal is in the first aorist passive voice/tense. This means the subject (you, which is really us) is being acted upon, and that it happened in the past and is a completed action. The spirit is in the instrumental case, which indicates it is the means by which this action was done upon us ("how or "by means of"), the Holy Spirit in essense, IS the seal.
The Scofield Bible has a useful footnote on this verse:
"The Holy Spirit is Himself the seal. In the symbolism of Scripture a seal signifies (1) A finished transaction (Jer. 32:9, 10; John 17:4, 19:30), (2) Ownership (Jer. 32:11, 12; 2 Tim. 2:19), (3) Security (Esth. 8:8; Dan. 6:17; Eph. 4:30)."
The sealing is completed, we are marked as property of God via it, and can be secure in that marking.