The Collaborative International Dictionary of English v.0.48:

Establish \Es*tab"lish\, v. t. [imp. & p. p. Established; p.
   pr. & vb. n. Establishing.] [OE. establissen, OF. establir,
   F. ['e]tablir, fr. L. stabilire, fr. stabilis firm, steady,
   stable. See Stable, a., -ish, and cf. Stablish.]
   1. To make stable or firm; to fix immovably or firmly; to set
      (a thing) in a place and make it stable there; to settle;
      to confirm.
      [1913 Webster]

            So were the churches established in the faith.
                                                  --Acts xvi. 5.
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            The best established tempers can scarcely forbear
            being borne down.                     --Burke.
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            Confidence which must precede union could be
            established only by consummate prudence and
            self-control.                         --Bancroft.
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   2. To appoint or constitute for permanence, as officers,
      laws, regulations, etc.; to enact; to ordain.
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            By the consent of all, we were established
            The people's magistrates.             --Shak.
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            Now, O king, establish the decree, and sign the
            writing, that it be not changed.      --Dan. vi. 8.
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   3. To originate and secure the permanent existence of; to
      found; to institute; to create and regulate; -- said of a
      colony, a state, or other institutions.
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            He hath established it [the earth], he created it
            not in vain, he formed it to be inhabited. --Is.
                                                  xlv. 18.
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            Woe to him that buildeth a town with blood, and
            establisheth a city by iniquity!      --Hab. ii. 12.
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   4. To secure public recognition in favor of; to prove and
      cause to be accepted as true; as, to establish a fact,
      usage, principle, opinion, doctrine, etc.
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            At the mouth of two witnesses, or at the mouth of
            three witnesses, shall the matter be established.
                                                  --Deut. xix.
                                                  15.
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   5. To set up in business; to place advantageously in a fixed
      condition; -- used reflexively; as, he established himself
      in a place; the enemy established themselves in the
      citadel.
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The Collaborative International Dictionary of English v.0.48:

established \established\ adj.
   1. brought about or set up or accepted; especially long and
      widely accepted; as, distrust of established authority; a
      team established as a member of a major league; enjoyed
      his prestige as an established writer; an established
      precedent; the established Church. Contrasted with
      unestablished. [Narrower terms: entrenched;
      implanted, planted, rooted; official; recognized]
      [WordNet 1.5]

   2. securely established; as, an established reputation.

   Syn: firm.
        [WordNet 1.5]

   3. settled securely and unconditionally.

   Syn: accomplished, effected.
        [WordNet 1.5]

   4. conforming with accepted standards.
      [WordNet 1.5]

   5. shown to be valid beyond a reasonable doubt; as, the
      established facts in the case.

   Syn: proved.
        [WordNet 1.5 +PJC]

   6. (Bot.) introduced from another region and persisting
      without cultivation; -- of plants.

   Syn: naturalized.
        [WordNet 1.5 +PJC]

WordNet (r) 3.0 (2006):

established
    adj 1: brought about or set up or accepted; especially long
           established; "the established social order"; "distrust
           the constituted authority"; "a team established as a
           member of a major league"; "enjoyed his prestige as an
           established writer"; "an established precedent"; "the
           established Church" [syn: established, constituted]
           [ant: unestablished]
    2: settled securely and unconditionally; "that smoking causes
       health problems is an accomplished fact" [syn:
       accomplished, effected, established]
    3: conforming with accepted standards; "a conventional view of
       the world" [syn: conventional, established]
    4: shown to be valid beyond a reasonable doubt; "the established
       facts in the case"
    5: introduced from another region and persisting without
       cultivation [syn: established, naturalized]